GCSE Maths: Tables of values and plotting quadratic graphs (Higher)
Part of the GCSE Maths quadratic and other graphs chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Tables of values and plotting quadratic graphs
In Chapter 12 you drew straight-line graphs. Some rules do not give a straight line. A rule that contains x² (“x squared”) gives a curve. The highest power of x is 2, and a rule like this is called a quadratic. Here are some quadratic rules: y = x², y = x² + 3, y = x² − 2x − 3 and y = 5 − x². The curve has its own name: a parabola.
To draw a quadratic graph we build up the picture from separate points. First we make a table of values. This lists some x-values and works out the matching y-value for each one by substituting: replacing x by the number. Each pair (x, y) is a coordinate. Then we plot the coordinates and join them with a smooth curve.
Be very careful with negative x-values. When you square a negative number the answer is positive, because a negative times a negative is positive: (−3)² = (−3) × (−3) = 9. On a calculator always put the negative number in brackets.
There are two possible shapes. If the number in front of x² is positive the curve is U-shaped and has a lowest point. If it is negative the curve is n-shaped (an upside-down U) and has a highest point. The lowest or highest point is called the turning point. The curve is perfectly symmetrical: it is a mirror image of itself, so the y-values in a table go down and then come back up in exactly the same order (for example 5, 0, −3, −4, −3, 0, 5).
Key ideas
- A quadratic has an x² term and no higher power. Its graph is a smooth curve called a parabola.
- Make a table of values: substitute each x into the rule to get y. Put negative numbers in brackets.
- Plot the points (x, y) and join them with a smooth freehand curve (no ruler, no pointed corners).
- Positive x² gives a U-shape; negative x² gives an n-shape.
- The curve is symmetrical about a vertical line through its turning point.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Put each x-value into the rule, one at a time, and work out y.
- Write the rule and the x-values. A table of values pairs every x-value with its y-value. Each y-value comes from putting that x-value into the rule.y = x2 + 2
x-values: −3, −2, −1, 0, 1, 2, 3 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −3: y = (−3)2 + 2 = 9 + 2 = 11
x = −2: y = (−2)2 + 2 = 4 + 2 = 6
x = −1: y = (−1)2 + 2 = 1 + 2 = 3
x = 0: y = 02 + 2 = 0 + 2 = 2
x = 1: y = 12 + 2 = 1 + 2 = 3
x = 2: y = 22 + 2 = 4 + 2 = 6
x = 3: y = 32 + 2 = 9 + 2 = 11 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: 11, 6, 3, 2, 3, 6, 11
Coordinates: (−3, 11), (−2, 6), (−1, 3), (0, 2), (1, 3), (2, 6), (3, 11)
Check. The gaps between neighbouring y-values are −5, −3, −1, 1, 3, 5. The gaps change by 2 every time (second differences are constant), which is what must happen for a quadratic. ✓
Watch out. Work out the x² part first. A negative number squared is positive.
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = −2: y = (−2)2 − 2 × (−2) − 3 = 4 + 4 − 3 = 5
x = −1: y = (−1)2 − 2 × (−1) − 3 = 1 + 2 − 3 = 0
x = 0: y = 02 − 2 × 0 − 3 = 0 + 0 − 3 = −3
x = 1: y = 12 − 2 × 1 − 3 = 1 − 2 − 3 = −4
x = 2: y = 22 − 2 × 2 − 3 = 4 − 4 − 3 = −3
x = 3: y = 32 − 2 × 3 − 3 = 9 − 6 − 3 = 0
x = 4: y = 42 − 2 × 4 − 3 = 16 − 8 − 3 = 5 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−2, 5), (−1, 0), (0, −3), (1, −4), (2, −3), (3, 0), (4, 5)
- Choose the scales for the axes. The x-values go from −2 to 4 and the y-values go from −4 to 5. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −2 to 4. y-axis: −5 to 6.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (−2, 5), (−1, 0), (0, −3), (1, −4), (2, −3), (3, 0), (4, 5).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, rounded at the turning point (no sharp corner).Label the curve y = x2 − 2x − 3.
Check. Every plotted point should lie on the curve, and the curve should pass through (−2, 5) and (4, 5) at the ends. It should be symmetrical about the line x = 1. ✓
Watch out. Do not use a ruler to join the points, and do not draw pointed corners. A quadratic is smooth and rounded at the turning point.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = −3: y = −(−3)2 + 5 = −9 + 5 = −4
x = −2: y = −(−2)2 + 5 = −4 + 5 = 1
x = −1: y = −(−1)2 + 5 = −1 + 5 = 4
x = 0: y = −(0)2 + 5 = 0 + 5 = 5
x = 1: y = −(1)2 + 5 = −1 + 5 = 4
x = 2: y = −(2)2 + 5 = −4 + 5 = 1
x = 3: y = −(3)2 + 5 = −9 + 5 = −4 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−3, −4), (−2, 1), (−1, 4), (0, 5), (1, 4), (2, 1), (3, −4)
- Choose the scales for the axes. The x-values go from −3 to 3 and the y-values go from −4 to 5. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −3 to 3. y-axis: −5 to 6.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (−3, −4), (−2, 1), (−1, 4), (0, 5), (1, 4), (2, 1), (3, −4).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, rounded at the turning point (no sharp corner).Label the curve y = −x2 + 5.
Check. Every plotted point should lie on the curve, and the curve should pass through (−3, −4) and (3, −4) at the ends. It should be symmetrical about the line x = 0. ✓
Watch out. Do not use a ruler to join the points, and do not draw pointed corners. A quadratic is smooth and rounded at the turning point.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. A point lies on a graph if its coordinates make the rule true. Substitute the x-value and see whether we get the y-value.
- Choose the x-value from the point. The point is (−1, 6), so x = −1 and the claim is that y = 6.y = x2 − x + 4
- Substitute x = −1 into the rule. Use brackets round the negative number. (−1)² = 1 and −(−1) = +1.y = (−1)2 − (−1) + 4 = 1 + 1 + 4 = 6
- Compare with the y-value in the point. We found y = 6, which is the same as the y-value in the point.y = 6, so the point (−1, 6) does lie on the graph.
Check. Try a different point: x = 0 gives y = 4, so (0, 4) is on the graph. The point (0, 6) would not be. ✓
Watch out. A point is on the graph only if the y-value matches exactly.
Watch out
Do not join the points with a ruler. A quadratic is a smooth curve, rounded at the turning point. A pointed bottom or straight pieces will lose marks. Also check every negative x: (−2)² is +4, not −4.Exam tip
In a table the first and last y-values are usually the biggest, and the table is symmetrical. If your values are not, check the negative x calculations again.Exercise 13A Tables of values and plotting quadratics
Non-calculatorKey words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and work out y, using BIDMAS.
- Write down the rule. The rule tells us how to get y from x. Here y comes from x2.y = x2
- Replace x by (−2). Substituting means putting the number in place of the letter. Put a negative number in brackets: (−2)² means (−2) × (−2) = 4.y = (−2)2
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = 4
- Write the answer. This tells us the curve passes through the point (−2, 4).When x = −2, y = 4
Check. Type it into a calculator using brackets for negatives. The display shows 4. ✓
Watch out. A negative number squared is positive: (−3)² = 9, not −9.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and work out y, using BIDMAS.
- Write down the rule. The rule tells us how to get y from x. Here y comes from x2 + 3.y = x2 + 3
- Replace x by (−1). Substituting means putting the number in place of the letter. Put a negative number in brackets: (−2)² means (−2) × (−2) = 4.y = (−1)2 + 3
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = 1 + 3 = 4
- Write the answer. This tells us the curve passes through the point (−1, 4).When x = −1, y = 4
Check. Type it into a calculator using brackets for negatives. The display shows 4. ✓
Watch out. A negative number squared is positive: (−3)² = 9, not −9.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and work out y, using BIDMAS.
- Write down the rule. The rule tells us how to get y from x. Here y comes from x2 − 5.y = x2 − 5
- Replace x by 3. Substituting means putting the number in place of the letter. Put a negative number in brackets: (−2)² means (−2) × (−2) = 4.y = 32 − 5
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = 9 − 5 = 4
- Write the answer. This tells us the curve passes through the point (3, 4).When x = 3, y = 4
Check. Type it into a calculator using brackets for negatives. The display shows 4. ✓
Watch out. Work out the power first, then multiply, then add or subtract.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and work out y, using BIDMAS.
- Write down the rule. The rule tells us how to get y from x. Here y comes from 2x2.y = 2x2
- Replace x by (−2). Substituting means putting the number in place of the letter. Put a negative number in brackets: (−2)² means (−2) × (−2) = 4.y = 2 × (−2)2
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = 8
- Write the answer. This tells us the curve passes through the point (−2, 8).When x = −2, y = 8
Check. Type it into a calculator using brackets for negatives. The display shows 8. ✓
Watch out. A negative number squared is positive: (−3)² = 9, not −9.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and work out y, using BIDMAS.
- Write down the rule. The rule tells us how to get y from x. Here y comes from x2 + x.y = x2 + x
- Replace x by (−3). Substituting means putting the number in place of the letter. Put a negative number in brackets: (−2)² means (−2) × (−2) = 4.y = (−3)2 + (−3)
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = 9 − 3 = 6
- Write the answer. This tells us the curve passes through the point (−3, 6).When x = −3, y = 6
Check. Use the factorised form y = (x + 1)x: (−3 + 1) × (−3) = (−2) × (−3) = 6. ✓
Watch out. A negative number squared is positive: (−3)² = 9, not −9.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and work out y, using BIDMAS.
- Write down the rule. The rule tells us how to get y from x. Here y comes from x2 − 3x + 1.y = x2 − 3x + 1
- Replace x by (−2). Substituting means putting the number in place of the letter. Put a negative number in brackets: (−2)² means (−2) × (−2) = 4.y = (−2)2 − 3 × (−2) + 1
- Work it out in the right order. Powers first, then multiplying and dividing, then adding and subtracting (BIDMAS).y = 4 + 6 + 1 = 11
- Write the answer. This tells us the curve passes through the point (−2, 11).When x = −2, y = 11
Check. Type it into a calculator using brackets for negatives. The display shows 11. ✓
Watch out. A negative number squared is positive: (−3)² = 9, not −9.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 6 | 2 | 6 |
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 11 | 6 | 3 | 2 | 3 | 6 | 11 |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Put each x-value into the rule, one at a time, and work out y.
- Write the rule and the x-values. A table of values pairs every x-value with its y-value. Each y-value comes from putting that x-value into the rule.y = x2 + 2
x-values: −3, −2, −1, 0, 1, 2, 3 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −3: y = (−3)2 + 2 = 9 + 2 = 11
x = −2: y = (−2)2 + 2 = 4 + 2 = 6
x = −1: y = (−1)2 + 2 = 1 + 2 = 3
x = 0: y = 02 + 2 = 0 + 2 = 2
x = 1: y = 12 + 2 = 1 + 2 = 3
x = 2: y = 22 + 2 = 4 + 2 = 6
x = 3: y = 32 + 2 = 9 + 2 = 11 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: 11, 6, 3, 2, 3, 6, 11
Coordinates: (−3, 11), (−2, 6), (−1, 3), (0, 2), (1, 3), (2, 6), (3, 11)
Check. The gaps between neighbouring y-values are −5, −3, −1, 1, 3, 5. The gaps change by 2 every time (second differences are constant), which is what must happen for a quadratic. ✓
Watch out. Work out the x² part first. A negative number squared is positive.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 5 | 0 | −3 | −4 | −3 | 0 | 5 |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Put each x-value into the rule, one at a time, and work out y.
- Write the rule and the x-values. A table of values pairs every x-value with its y-value. Each y-value comes from putting that x-value into the rule.y = x2 − 4
x-values: −3, −2, −1, 0, 1, 2, 3 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −3: y = (−3)2 − 4 = 9 − 4 = 5
x = −2: y = (−2)2 − 4 = 4 − 4 = 0
x = −1: y = (−1)2 − 4 = 1 − 4 = −3
x = 0: y = 02 − 4 = 0 − 4 = −4
x = 1: y = 12 − 4 = 1 − 4 = −3
x = 2: y = 22 − 4 = 4 − 4 = 0
x = 3: y = 32 − 4 = 9 − 4 = 5 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: 5, 0, −3, −4, −3, 0, 5
Coordinates: (−3, 5), (−2, 0), (−1, −3), (0, −4), (1, −3), (2, 0), (3, 5)
Check. The gaps between neighbouring y-values are −5, −3, −1, 1, 3, 5. The gaps change by 2 every time (second differences are constant), which is what must happen for a quadratic. ✓
Watch out. Work out the x² part first. A negative number squared is positive.
| x | −3 | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|---|
| y |
| x | −3 | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|---|
| y | 6 | 2 | 0 | 0 | 2 | 6 |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Put each x-value into the rule, one at a time, and work out y.
- Write the rule and the x-values. A table of values pairs every x-value with its y-value. Each y-value comes from putting that x-value into the rule.y = x2 + x
x-values: −3, −2, −1, 0, 1, 2 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = −3: y = (−3)2 + (−3) = 9 − 3 = 6
x = −2: y = (−2)2 + (−2) = 4 − 2 = 2
x = −1: y = (−1)2 + (−1) = 1 − 1 = 0
x = 0: y = 02 + 0 = 0 + 0 = 0
x = 1: y = 12 + 1 = 1 + 1 = 2
x = 2: y = 22 + 2 = 4 + 2 = 6 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: 6, 2, 0, 0, 2, 6
Coordinates: (−3, 6), (−2, 2), (−1, 0), (0, 0), (1, 2), (2, 6)
Check. The gaps between neighbouring y-values are −4, −2, 0, 2, 4. The gaps change by 2 every time (second differences are constant), which is what must happen for a quadratic. ✓
Watch out. Work out the x² part first. A negative number squared is positive.
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = −3: y = (−3)2 + 1 = 9 + 1 = 10
x = −2: y = (−2)2 + 1 = 4 + 1 = 5
x = −1: y = (−1)2 + 1 = 1 + 1 = 2
x = 0: y = 02 + 1 = 0 + 1 = 1
x = 1: y = 12 + 1 = 1 + 1 = 2
x = 2: y = 22 + 1 = 4 + 1 = 5
x = 3: y = 32 + 1 = 9 + 1 = 10 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−3, 10), (−2, 5), (−1, 2), (0, 1), (1, 2), (2, 5), (3, 10)
- Choose the scales for the axes. The x-values go from −3 to 3 and the y-values go from 1 to 10. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −3 to 3. y-axis: 0 to 11.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (−3, 10), (−2, 5), (−1, 2), (0, 1), (1, 2), (2, 5), (3, 10).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, rounded at the turning point (no sharp corner).Label the curve y = x2 + 1.
Check. Every plotted point should lie on the curve, and the curve should pass through (−3, 10) and (3, 10) at the ends. It should be symmetrical about the line x = 0. ✓
Watch out. Do not use a ruler to join the points, and do not draw pointed corners. A quadratic is smooth and rounded at the turning point.
| x | −1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|
| y |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = −1: y = (−1)2 − 4 × (−1) + 3 = 1 + 4 + 3 = 8
x = 0: y = 02 − 4 × 0 + 3 = 0 + 0 + 3 = 3
x = 1: y = 12 − 4 × 1 + 3 = 1 − 4 + 3 = 0
x = 2: y = 22 − 4 × 2 + 3 = 4 − 8 + 3 = −1
x = 3: y = 32 − 4 × 3 + 3 = 9 − 12 + 3 = 0
x = 4: y = 42 − 4 × 4 + 3 = 16 − 16 + 3 = 3
x = 5: y = 52 − 4 × 5 + 3 = 25 − 20 + 3 = 8 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−1, 8), (0, 3), (1, 0), (2, −1), (3, 0), (4, 3), (5, 8)
- Choose the scales for the axes. The x-values go from −1 to 5 and the y-values go from −1 to 8. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −1 to 5. y-axis: −2 to 9.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (−1, 8), (0, 3), (1, 0), (2, −1), (3, 0), (4, 3), (5, 8).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, rounded at the turning point (no sharp corner).Label the curve y = x2 − 4x + 3.
Check. Every plotted point should lie on the curve, and the curve should pass through (−1, 8) and (5, 8) at the ends. It should be symmetrical about the line x = 2. ✓
Watch out. Do not use a ruler to join the points, and do not draw pointed corners. A quadratic is smooth and rounded at the turning point.
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|
| y |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = −2: y = −(−2)2 + 2 × (−2) + 3 = −4 − 4 + 3 = −5
x = −1: y = −(−1)2 + 2 × (−1) + 3 = −1 − 2 + 3 = 0
x = 0: y = −(0)2 + 2 × 0 + 3 = 0 + 0 + 3 = 3
x = 1: y = −(1)2 + 2 × 1 + 3 = −1 + 2 + 3 = 4
x = 2: y = −(2)2 + 2 × 2 + 3 = −4 + 4 + 3 = 3
x = 3: y = −(3)2 + 2 × 3 + 3 = −9 + 6 + 3 = 0
x = 4: y = −(4)2 + 2 × 4 + 3 = −16 + 8 + 3 = −5 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−2, −5), (−1, 0), (0, 3), (1, 4), (2, 3), (3, 0), (4, −5)
- Choose the scales for the axes. The x-values go from −2 to 4 and the y-values go from −5 to 4. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −2 to 4. y-axis: −6 to 5.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (−2, −5), (−1, 0), (0, 3), (1, 4), (2, 3), (3, 0), (4, −5).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, rounded at the turning point (no sharp corner).Label the curve y = −x2 + 2x + 3.
Check. Every plotted point should lie on the curve, and the curve should pass through (−2, −5) and (4, −5) at the ends. It should be symmetrical about the line x = 1. ✓
Watch out. Do not use a ruler to join the points, and do not draw pointed corners. A quadratic is smooth and rounded at the turning point.
Method. Find the wrong step in her working, then redo the calculation correctly.
- Look at her squaring step. Squaring means multiplying a number by itself: (−3)² = (−3) × (−3).(−3) × (−3) = 9, not −9
- Explain the rule. A negative number multiplied by a negative number gives a positive number. So the square of any number is positive (or zero).Her mistake: she gave a negative answer for (−3)².
- Redo the calculation. Use the correct value 9, then add 2.y = 9 + 2 = 11
Check. Using the point (3, 11): y(3) = 9 + 2 = 11, the same as y(−3), as the symmetry of the curve says. ✓
Watch out. If you ever get a negative answer for a square, there is a mistake.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and compare the result with the y-value of the point.
- Write the rule and the point. The point is (2, 1), so x = 2 and we need to see whether y = 1.y = x2 − 3
- Substitute x. Put negative numbers in brackets.y = 22 − 3 = 4 − 3 = 1
- Compare. The rule gives y = 1. The point has y = 1.1 = 1, so the point is on the graph.
Check. A point on this curve with x = 2 must have y = 1; (2, 1) is the point on the curve. ✓
Watch out. Only an exact match means the point is on the curve.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and compare the result with the y-value of the point.
- Write the rule and the point. The point is (−1, −4), so x = −1 and we need to see whether y = −4.y = x2 − 3
- Substitute x. Put negative numbers in brackets.y = (−1)2 − 3 = 1 − 3 = −2
- Compare. The rule gives y = −2. The point has y = −4.−2 ≠ −4, so the point is not on the graph.
Check. A point on this curve with x = −1 must have y = −2; (−1, −2) is the point on the curve. ✓
Watch out. Only an exact match means the point is on the curve.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and compare the result with the y-value of the point.
- Write the rule and the point. The point is (0, −3), so x = 0 and we need to see whether y = −3.y = x2 − 3
- Substitute x. Put negative numbers in brackets.y = 02 − 3 = 0 − 3 = −3
- Compare. The rule gives y = −3. The point has y = −3.−3 = −3, so the point is on the graph.
Check. A point on this curve with x = 0 must have y = −3; (0, −3) is the point on the curve. ✓
Watch out. Only an exact match means the point is on the curve.
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Substitute the x-value into the rule and compare the result with the y-value of the point.
- Write the rule and the point. The point is (−3, 6), so x = −3 and we need to see whether y = 6.y = x2 − 3
- Substitute x. Put negative numbers in brackets.y = (−3)2 − 3 = 9 − 3 = 6
- Compare. The rule gives y = 6. The point has y = 6.6 = 6, so the point is on the graph.
Check. A point on this curve with x = −3 must have y = 6; (−3, 6) is the point on the curve. ✓
Watch out. Only an exact match means the point is on the curve.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The sign of the x² term decides the shape.
- Look at the overall shape. A quadratic graph is a parabola. If the curve opens upwards it is U-shaped; if it opens downwards it is n-shaped.The curve opens upwards.
- Say what that means for the x² term. A positive number in front of x² gives a U-shape. A negative number gives an n-shape.The number in front of x2 is 1, which is positive.
Check. Put x = 10 and x = −10 into the rule: y is large and positive both times. ✓
Watch out. Do not confuse the shape with the turning point: U-shaped has a minimum, n-shaped has a maximum.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The sign of the x² term decides the shape.
- Look at the overall shape. A quadratic graph is a parabola. If the curve opens upwards it is U-shaped; if it opens downwards it is n-shaped.The curve opens downwards.
- Say what that means for the x² term. A positive number in front of x² gives a U-shape. A negative number gives an n-shape.The number in front of x2 is −1, which is negative.
Check. Put x = 10 and x = −10 into the rule: y is large and negative both times. ✓
Watch out. Do not confuse the shape with the turning point: U-shaped has a minimum, n-shaped has a maximum.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The sign of the x² term decides the shape.
- Look at the overall shape. A quadratic graph is a parabola. If the curve opens upwards it is U-shaped; if it opens downwards it is n-shaped.The curve opens downwards.
- Say what that means for the x² term. A positive number in front of x² gives a U-shape. A negative number gives an n-shape.The number in front of x2 is −1, which is negative.
Check. Put x = 10 and x = −10 into the rule: y is large and negative both times. ✓
Watch out. Do not confuse the shape with the turning point: U-shaped has a minimum, n-shaped has a maximum.
Key words. A quadratic is a rule where the highest power of x is x² (x squared). Its graph is a smooth curve called a parabola. The turning point is where the curve changes direction (the lowest or highest point). A root is an x-value where the curve meets the x-axis (where y = 0). The y-intercept is where the curve crosses the y-axis (where x = 0).
Method. The sign of the x² term decides the shape.
- Look at the overall shape. A quadratic graph is a parabola. If the curve opens upwards it is U-shaped; if it opens downwards it is n-shaped.The curve opens upwards.
- Say what that means for the x² term. A positive number in front of x² gives a U-shape. A negative number gives an n-shape.The number in front of x2 is 1, which is positive.
Check. Put x = 10 and x = −10 into the rule: y is large and positive both times. ✓
Watch out. Do not confuse the shape with the turning point: U-shaped has a minimum, n-shaped has a maximum.
Method. Put y = 11 into the rule to get an equation for k. Then solve it, remembering that squaring can give two answers.
- Write the rule with the point. The point (k, 11) has x = k and y = 11.11 = k² + 2
- Subtract 2 from both sides. We want k² on its own.k² = 11 − 2 = 9
- Take the square root. Two numbers square to 9: 3 × 3 = 9 and (−3) × (−3) = 9.k = 3 or k = −3
- Write the answer. The graph is symmetrical, so both (3, 11) and (−3, 11) are on it.k = 3 or k = −3
Check. x = 3: 9 + 2 = 11 ✓. x = −3: (−3)² + 2 = 9 + 2 = 11 ✓.
Watch out. Do not forget the negative square root.
| x | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| y |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = −2: y = 2 × (−2)2 − 3 × (−2) − 5 = 8 + 6 − 5 = 9
x = −1: y = 2 × (−1)2 − 3 × (−1) − 5 = 2 + 3 − 5 = 0
x = 0: y = 2 × 02 − 3 × 0 − 5 = 0 + 0 − 5 = −5
x = 1: y = 2 × 12 − 3 × 1 − 5 = 2 − 3 − 5 = −6
x = 2: y = 2 × 22 − 3 × 2 − 5 = 8 − 6 − 5 = −3
x = 3: y = 2 × 32 − 3 × 3 − 5 = 18 − 9 − 5 = 4 - List the coordinates. Each x with its y makes a point (x, y) to plot.(−2, 9), (−1, 0), (0, −5), (1, −6), (2, −3), (3, 4)
- Choose the scales for the axes. The x-values go from −2 to 3 and the y-values go from −6 to 9. The axes must cover all of these, and each square must always stand for the same amount.x-axis: −2 to 3. y-axis: −8 to 12.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (−2, 9), (−1, 0), (0, −5), (1, −6), (2, −3), (3, 4).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, rounded at the turning point (no sharp corner).Label the curve y = 2x2 − 3x − 5.
Check. Every plotted point should lie on the curve, and the curve should pass through (−2, 9) and (3, 4) at the ends. It should be symmetrical about the line x = 0.75. ✓
Watch out. Do not use a ruler to join the points, and do not draw pointed corners. A quadratic is smooth and rounded at the turning point.
| x | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|---|---|
| y |
Turning point (1.5, −2.25).
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Make a table of values, plot the points, then join them with a smooth curve.
- Make a table of values. Each y-value comes from putting an x-value into the rule. Use brackets round negative numbers.x = 0: y = 02 − 3 × 0 = 0 + 0 = 0
x = 0.5: y = 0.52 − 3 × 0.5 = 0.25 − 1.5 = −1.25
x = 1: y = 12 − 3 × 1 = 1 − 3 = −2
x = 1.5: y = 1.52 − 3 × 1.5 = 2.25 − 4.5 = −2.25
x = 2: y = 22 − 3 × 2 = 4 − 6 = −2
x = 2.5: y = 2.52 − 3 × 2.5 = 6.25 − 7.5 = −1.25
x = 3: y = 32 − 3 × 3 = 9 − 9 = 0 - List the coordinates. Each x with its y makes a point (x, y) to plot.(0, 0), (0.5, −1.25), (1, −2), (1.5, −2.25), (2, −2), (2.5, −1.25), (3, 0)
- Choose the scales for the axes. The x-values go from 0 to 3 and the y-values go from −2.25 to 0. The axes must cover all of these, and each square must always stand for the same amount.x-axis: 0 to 3. y-axis: −4 to 1.
- Plot each point carefully. Go along the x-axis first, then up or down to the y-value. Mark each point with a small cross.Plot (0, 0), (0.5, −1.25), (1, −2), (1.5, −2.25), (2, −2), (2.5, −1.25), (3, 0).
- Join the points with a smooth curve. The graph of this rule is a smooth curve, not a set of straight lines. Draw it freehand in one flowing line, rounded at the turning point (no sharp corner).Label the curve y = x2 − 3x.
Check. Every plotted point should lie on the curve, and the curve should pass through (0, 0) and (3, 0) at the ends. It should be symmetrical about the line x = 1.5. ✓
Watch out. Do not use a ruler to join the points, and do not draw pointed corners. A quadratic is smooth and rounded at the turning point.
Method. Think about the sign of x² for every possible x-value.
- Positive x. A positive number times a positive number is positive.3 × 3 = 9, which is positive
- Negative x. A negative times a negative is positive.(−3) × (−3) = 9, which is positive
- Zero. 0 × 0 = 0.y = 0
- Conclude. y = x² is never negative, so the curve is never below the x-axis (y = 0). It touches the axis only at (0, 0).y is always 0 or positive.
Check. Try x = −4, −0.5, 5: y = 16, 0.25, 25. All positive. ✓
Watch out. This is why every table for y = x² has its smallest value at x = 0.
Method. Think about what the graph of a quadratic looks like between the plotted points.
- Recall the shape. The graph of a quadratic is a parabola: a smooth curve with no straight parts and no corners.
- Compare with Ben's drawing. A ruler gives straight sections with pointed corners at each plotted point. The real graph bends smoothly between the points and is rounded at the turning point.
- Say how to fix it. Plot more points if needed, then draw one smooth curve freehand through them.Ben should draw a smooth curve.
Check. Test x = 0.5: the rule gives y = 0.25 − 1 − 3 = −3.75, but the ruler line between (0, −3) and (1, −4) is at −3.5. The true curve is lower, so the ruler line is wrong. ✓
Watch out. Quadratic graphs have no straight parts.
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| y |
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| y | 8 | 3 | 0 | −1 | 0 | 3 | 8 |
Key words. To substitute means to replace the letter by a number. A table of values lists x-values and the y-values that go with them. Each pair (x, y) is a coordinate that can be plotted.
Method. Put each x-value into the rule, one at a time, and work out y.
- Write the rule and the x-values. A table of values pairs every x-value with its y-value. Each y-value comes from putting that x-value into the rule.y = x2 − 6x + 8
x-values: 0, 1, 2, 3, 4, 5, 6 - Substitute the x-values one at a time. Do the same calculation every time, only the number changes. Use brackets round negative numbers.x = 0: y = 02 − 6 × 0 + 8 = 0 + 0 + 8 = 8
x = 1: y = 12 − 6 × 1 + 8 = 1 − 6 + 8 = 3
x = 2: y = 22 − 6 × 2 + 8 = 4 − 12 + 8 = 0
x = 3: y = 32 − 6 × 3 + 8 = 9 − 18 + 8 = −1
x = 4: y = 42 − 6 × 4 + 8 = 16 − 24 + 8 = 0
x = 5: y = 52 − 6 × 5 + 8 = 25 − 30 + 8 = 3
x = 6: y = 62 − 6 × 6 + 8 = 36 − 36 + 8 = 8 - Write the y-values in the table. Each pair of numbers is a coordinate, ready to be plotted on a graph.y-values: 8, 3, 0, −1, 0, 3, 8
Coordinates: (0, 8), (1, 3), (2, 0), (3, −1), (4, 0), (5, 3), (6, 8)
Check. The gaps between neighbouring y-values are −5, −3, −1, 1, 3, 5. The gaps change by 2 every time (second differences are constant), which is what must happen for a quadratic. ✓
Watch out. Work out the x² part first. A negative number squared is positive.
Method. A quadratic table is symmetrical, so equal y-values come in pairs either side of the line of symmetry.
- Find y = 3 in the table. The table is 8, 3, 0, −1, 0, 3, 8.y = 3 when x = 1 (given)
- Look for the other 3. It appears again on the other side of the lowest value.The y-values are 8, 3, 0, −1, 0, 3, 8, so the other 3 is at x = 5
Check. x = 5: y = 52 − 6 × 5 + 8 = 25 − 30 + 8 = 3 ✓
Watch out. Look for repeated y-values.
Method. The line of symmetry is halfway between two matching points.
- Take a matching pair. The points (1, 3) and (5, 3) are mirror images.x = 1 and x = 5
- Find the midpoint of the x-values. Halfway between 1 and 5.(1 + 5) ÷ 2 = 3
- Write the vertical line. A vertical line has equation x = number.x = 3
Check. The lowest value −1 is at x = 3, in the middle of the table. ✓
Watch out. Vertical lines are x = number.
Method. Explain using the pairs in the table.
- Look at the pairs. x = 2 and x = 4 both give 0. x = 1 and x = 5 both give 3. x = 0 and x = 6 both give 8.
- Describe the pattern. Each pair is the same distance either side of x = 3, and has the same y-value.So the graph is a mirror image about x = 3.
Check. The lowest point is at x = 3, directly between all the pairs. ✓
Watch out. Give a reason, not just the answer.