Quadratic basics

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   (−3) × (−3)
2. Work it out. Negative times negative gives a positive answer.
   9
3. Write the answer.
   9

Check. The square is never negative, because two negatives multiply to a positive.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   (−4) × (−4)
2. Work it out. Negative times negative gives a positive answer.
   16
3. Write the answer.
   16

Check. The square is never negative, because two negatives multiply to a positive.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   5 × 5
2. Work it out. Negative times negative gives a positive answer.
   25
3. Write the answer.
   25

Check. 5 × 5 = 25.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   (−6) × (−6)
2. Work it out. Negative times negative gives a positive answer.
   36
3. Write the answer.
   36

Check. The square is never negative, because two negatives multiply to a positive.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   7 × 7
2. Work it out. Negative times negative gives a positive answer.
   49
3. Write the answer.
   49

Check. 7 × 7 = 49.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   (−8) × (−8)
2. Work it out. Negative times negative gives a positive answer.
   64
3. Write the answer.
   64

Check. The square is never negative, because two negatives multiply to a positive.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   9 × 9
2. Work it out. Negative times negative gives a positive answer.
   81
3. Write the answer.
   81

Check. 9 × 9 = 81.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   10 × 10
2. Work it out. Negative times negative gives a positive answer.
   100
3. Write the answer.
   100

Check. 10 × 10 = 100.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Without brackets, the power applies to 5 only.

1. Find 5². Five squared is 25.
   5 × 5 = 25
2. Apply the minus sign. The minus sign is applied to the answer.
   −(25)
3. Write the answer.
   −25

Check. (−5)² = 25 is different.

Method. Square the negative number, using the brackets.

1. Write the square. (−4) × (−4).
   (−4) × (−4)
2. Work it out. A negative times a negative is positive.
   16
3. Write the answer. Ali is not right, because (−4)² = 16.
   No

Check. −16 would be −(4²), which is a different calculation.

Method. Put x = 6 into x².

1. Put x = 6 into the rule. The distance is x², so use 6².
   6²
2. Work it out. Six squared is 36.
   6 × 6 = 36
3. Write the answer.
   36 m

Check. The ball falls more each second, so after 6 seconds the distance is larger than after 3 seconds (9 m).

Method. Square first, using the brackets, then add 3.

1. Square −4. Put x = −4 into x². The square is positive.
   (−4)² = 16
2. Add 3. Add 3 to the square.
   16 + 3 = 19

Check. 19 − 3 = 16, and 16 is the square of −4.

Method. Square first, then multiply by 2.

1. Square −3. (−3)² = 9.
   9
2. Multiply by 2. The 2 in front multiplies the square.
   2 × 9 = 18

Check. 18 ÷ 2 = 9.

Method. Square each value. Remember that a negative times a negative is positive.

1. Square −2 and −1. (−2)² = 4 and (−1)² = 1.
   4 and 1
2. Square 0, 1 and 2. 0² = 0, 1² = 1 and 2² = 4.
   0, 1 and 4
3. Write the answer. The values are the same for −1 and 1, and for −2 and 2.
   4, 1, 0, 1, 4

Check. The squares are symmetrical about 0.

Method. Square each number, then compare.

1. Work out (−5)². A negative times a negative is positive.
   25
2. Work out 5². Five squared is 25.
   25
3. Compare. The two answers are equal.
   25 = 25
4. Write the answer.
   They are equal. Both are 25.

Check. The square of a negative and the square of the positive number give the same answer.

Method. Square first, multiply by 3, then subtract 5.

1. Square 2. 2² = 4.
   4
2. Multiply by 3. 3 × 4 = 12.
   12
3. Subtract 5. 12 − 5 = 7.
   7

Check. 7 + 5 = 12, and 12 ÷ 3 = 4.

Method. Square each x value, multiply by 1 if needed, then add 1.

1. Find x² for x = -2. Square the number first.
   −2² = 4
2. Find x² for x = -1. Square the number first.
   −1² = 1
3. Find x² for x = 0. Square the number first.
   0² = 0
4. Work out y for each x. Multiply x² by 1 (if needed), then add 1.
   4 + 1 = 5, 1 + 1 = 2, 0 + 1 = 1
5. Write the table. Each y value goes with its x value.
   (-2, 5), (-1, 2), (0, 1), (1, 2), (2, 5)

Check. Check one value: when x = -2, y = x² + 1 gives 5.

Method. Square each x value, multiply by 1 if needed, then add -2.

1. Find x² for x = -2. Square the number first.
   −2² = 4
2. Find x² for x = -1. Square the number first.
   −1² = 1
3. Find x² for x = 0. Square the number first.
   0² = 0
4. Work out y for each x. Multiply x² by 1 (if needed), then add -2.
   4 − 2 = 2, 1 − 2 = -1, 0 − 2 = -2
5. Write the table. Each y value goes with its x value.
   (-2, 2), (-1, -1), (0, -2), (1, -1), (2, 2)

Check. Check one value: when x = -2, y = x² − 2 gives 2.

Method. Square each x value, multiply by 1 if needed, then add 3.

1. Find x² for x = -3. Square the number first.
   −3² = 9
2. Find x² for x = -2. Square the number first.
   −2² = 4
3. Find x² for x = -1. Square the number first.
   −1² = 1
4. Work out y for each x. Multiply x² by 1 (if needed), then add 3.
   9 + 3 = 12, 4 + 3 = 7, 1 + 3 = 4
5. Write the table. Each y value goes with its x value.
   (-3, 12), (-2, 7), (-1, 4), (0, 3), (1, 4), (2, 7), (3, 12)

Check. Check one value: when x = -3, y = x² + 3 gives 12.

Method. Square each x value, multiply by 1 if needed, then add -4.

1. Find x² for x = -3. Square the number first.
   −3² = 9
2. Find x² for x = -2. Square the number first.
   −2² = 4
3. Find x² for x = -1. Square the number first.
   −1² = 1
4. Work out y for each x. Multiply x² by 1 (if needed), then add -4.
   9 − 4 = 5, 4 − 4 = 0, 1 − 4 = -3
5. Write the table. Each y value goes with its x value.
   (-3, 5), (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0), (3, 5)

Check. Check one value: when x = -3, y = x² − 4 gives 5.

Method. Use the table, plot each point, then join them with a smooth curve.

1. Use the table. The points are (−2, 2), (−1, −1), (0, −2), (1, −1) and (2, 2).
   5 points
2. Plot the points. Plot each point on the grid.
   five points
3. Join with a smooth curve. Do not join with straight lines. The curve is U-shaped.
   smooth U-shaped curve

Check. The lowest point is at (0, −2), on the line of symmetry x = 0.

Method. Put y = 7 and solve x² − 2 = 7.

1. Write the equation. Put y = 7 into the rule.
   x² − 2 = 7
2. Add 2 to both sides. This moves the number to the other side.
   x² = 9
3. Take the square root. Both 3 and −3 square to give 9.
   x = 3 or x = −3
4. Write the answer.
   x = 3 or x = −3

Check. 3² − 2 = 7, and (−3)² − 2 = 7.

Method. Put x = 2 into the rule and compare with y = 5.

1. Put x = 2 into y = x² + 1. 2² + 1 = 4 + 1.
   4 + 1 = 5
2. Compare. The point has y = 5, which matches.
   5 = 5
3. Write the answer.
   Yes

Check. (2, 5) satisfies y = x² + 1.

Method. Put x = 0 into the rule.

1. Put x = 0 into y = x² − 4. 0² − 4 = 0 − 4 = −4.
   −4
2. Compare. The y value should be −4, not 4.
   −4 ≠ 4
3. Write the answer. Ali is not right. The point is (0, −4).
   No

Check. The graph of y = x² − 4 crosses the y-axis at (0, −4).

Method. Square each x value.

1. Square 2. 2² = 4.
   4
2. Square −2. (−2)² = 4.
   4
3. Add 3. Both give y = 4 + 3 = 7.
   7
4. Write the answer. The graph is symmetrical, so x = 2 and x = −2 give the same height.
   Both give 7

Check. The two x values are equal in size and opposite in sign.

Method. Square each x value, multiply by 2 if needed, then add 1.

1. Find x² for x = -2. Square the number first.
   −2² = 4
2. Find x² for x = -1. Square the number first.
   −1² = 1
3. Find x² for x = 0. Square the number first.
   0² = 0
4. Work out y for each x. Multiply x² by 2 (if needed), then add 1.
   2 × 4 + 1 = 9, 2 × 1 + 1 = 3, 2 × 0 + 1 = 1
5. Write the table. Each y value goes with its x value.
   (-2, 9), (-1, 3), (0, 1), (1, 3), (2, 9)

Check. Check one value: when x = -2, y = 2x² + 1 gives 9.

Method. Square each x value, multiply by -1 if needed, then add 4.

1. Find x² for x = -3. Square the number first.
   −3² = 9
2. Find x² for x = -2. Square the number first.
   −2² = 4
3. Find x² for x = -1. Square the number first.
   −1² = 1
4. Work out y for each x. Multiply x² by -1 (if needed), then add 4.
   −1 × 9 + 4 = -5, −1 × 4 + 4 = 0, −1 × 1 + 4 = 3
5. Write the table. Each y value goes with its x value.
   (-3, -5), (-2, 0), (-1, 3), (0, 4), (1, 3), (2, 0), (3, -5)

Check. Check one value: when x = -3, y = −x² + 4 gives -5.

Method. Square each x value, multiply by 3 if needed, then add -2.

1. Find x² for x = -2. Square the number first.
   −2² = 4
2. Find x² for x = -1. Square the number first.
   −1² = 1
3. Find x² for x = 0. Square the number first.
   0² = 0
4. Work out y for each x. Multiply x² by 3 (if needed), then add -2.
   3 × 4 − 2 = 10, 3 × 1 − 2 = 1, 3 × 0 − 2 = -2
5. Write the table. Each y value goes with its x value.
   (-2, 10), (-1, 1), (0, -2), (1, 1), (2, 10)

Check. Check one value: when x = -2, y = 3x² − 2 gives 10.

Method. A larger value of a makes the curve narrower, because the y values rise faster.

1. Compare the y values. At x = 1: y = 1 + 1 = 2 for the first, and 3 + 1 = 4 for the second.
   2 and 4
2. Decide. The second curve rises faster, so it is narrower.
   a = 3 is bigger than a = 1
3. Write the answer.
   y = 3x² + 1

Check. The larger a gives steeper sides.

Method. A negative value of a turns the curve upside down.

1. Look at the sign of a. Negative a means the curve opens downwards.
   a = −2 is negative
2. Write the answer. The first curve opens upwards, because a = 2 is positive.
   y = −2x² + 3

Check. The table of values for −2x² + 3 goes down in the middle.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 + 4.
   y = 4
3. Write the answer.
   (0, 4)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 − 6.
   y = -6
3. Write the answer.
   (0, -6)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. Moving the graph up by 5 adds 5 to the y value.

1. Start with the original equation. The original is y = x² − 3.
   c = −3
2. Add 5 to c. Moving the graph up 5 units adds 5 to every y value.
   −3 + 5 = 2
3. Write the answer.
   y = x² + 2

Check. The turning point moves from (0, −3) to (0, 2).

Method. The highest point of y = −x² + c is the turning point at (0, c).

1. Find the x value. The x² term is zero when x = 0, which is the highest point.
   x = 0
2. Find the y value. Put x = 0 into the equation: y = −0 + 2.
   y = 2
3. Write the answer.
   (0, 2)

Check. Any other x makes −x² smaller, so 2 is the highest value.

Method. Decide whether the curve opens up or down.

1. Look at a. a = −1 is negative, so the curve is upside down.
   a < 0
2. Find the turning point. The turning point is at (0, 4), which is the highest point, not the lowest.
   (0, 4) is the highest point
3. Write the answer. Ali is not right.
   No

Check. y = −x² + 4 gets smaller away from x = 0.

Method. Compare the turning points.

1. Find the turning points. For c = −1: (0, −1). For c = 2: (0, 2).
   (0, −1) and (0, 2)
2. Compare. The second curve is 3 units higher.
   2 − (−1) = 3
3. Write the answer.
   The second curve is 3 units higher.

Check. The same shape, but shifted.

Method. Subtract the y values of the two turning points.

1. Find the difference. Subtract the smaller y value from the larger.
   5 − (−1) = 6
2. Write the answer.
   6

Check. −1 + 6 = 5.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 4 = 0
2. Rearrange. Move c to the other side.
   x² = 4
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±2
4. Write the answer.
   x = 2 or x = −2

Check. Check: 2² = 4 and (−2)² = 4. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 9 = 0
2. Rearrange. Move c to the other side.
   x² = 9
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±3
4. Write the answer.
   x = 3 or x = −3

Check. Check: 3² = 9 and (−3)² = 9. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 16 = 0
2. Rearrange. Move c to the other side.
   x² = 16
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±4
4. Write the answer.
   x = 4 or x = −4

Check. Check: 4² = 16 and (−4)² = 16. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 1 = 0
2. Rearrange. Move c to the other side.
   x² = 1
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±1
4. Write the answer.
   x = 1 or x = −1

Check. Check: 1² = 1 and (−1)² = 1. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. x² is never negative, so x² + c = 0 has no solutions when c is positive.

1. Write the equation. Set y = 0.
   x² + 5 = 0
2. Rearrange. Move c to the other side.
   x² = -5
3. Decide. x² cannot be negative, so there is no real solution.
   no real solutions
4. Write the answer.
   No solutions

Check. The graph is above the x-axis, so it never crosses it.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 2 = 0
2. Rearrange. Move c to the other side.
   x² = 2
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±√2 ≈ 1.41
4. Write the answer.
   x ≈ −1.4 or x ≈ 1.4

Check. √2 is about 1.41, and both answers are about 1.4 from zero.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. Solve x² = 9, then check.

1. Write the equation. Set y = 0, so x² − 9 = 0.
   x² = 9
2. Take the square root. The roots are x = 3 and x = −3.
   x = ±3
3. Compare. Sam wrote 9 and −9, which are not the roots.
   9 ≠ 3
4. Write the answer. Sam is not right. The roots are 3 and −3.
   No

Check. 3² = 9, so x = 3 gives x² = 9, not x = 9.

Method. x² is never negative, so x² + c = 0 has no solutions when c is positive.

1. Write the equation. Set y = 0.
   x² + 9 = 0
2. Rearrange. Move c to the other side.
   x² = -9
3. Decide. x² cannot be negative, so there is no real solution.
   no real solutions
4. Write the answer.
   No solutions

Check. The graph is above the x-axis, so it never crosses it.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 16 = 0
2. Rearrange. Move c to the other side.
   x² = 16
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±4
4. Write the answer.
   x = 4 or x = −4

Check. Check: 4² = 16 and (−4)² = 16. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 1 = 0
2. Rearrange. Move c to the other side.
   x² = 1
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±1
4. Write the answer.
   x = 1 or x = −1

Check. Check: 1² = 1 and (−1)² = 1. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. The roots are the x values where y = 0. Write each root as a coordinate with y = 0.

1. Find the roots. Solve x² − 4 = 0.
   x = ±2
2. Write as coordinates. Each point has y = 0.
   (2, 0) and (−2, 0)
3. Write the answer.
   (2, 0) and (−2, 0)

Check. Both points lie on the x-axis.

Method. Use the roots to find c. At a root y = 0, so x² + c = 0.

1. Use x = 3. At x = 3, y = 0, so 3² + c = 0.
   9 + c = 0
2. Solve for c. Take 9 from both sides.
   c = −9
3. Write the equation.
   y = x² − 9

Check. At x = −3: (−3)² − 9 = 0.

Method. Solve x² = 6.

1. Write the equation. Set y = 0: x² = 6.
   x² = 6
2. Take the square root. √6 is about 2.45, so the roots are about ±2.4.
   x ≈ ±2.4
3. Compare. Ali said ±6, which is not right.
   6 ≠ 2.4
4. Write the answer. Ali is not right.
   No

Check. 2.4² is about 5.8, close to 6.

Method. A quadratic has x² as its highest power.

1. Look at each expression. x² + 3x has highest power 2.
   yes
2. Look at the second. 3x + 2 has highest power 1, so it is not quadratic.
   no
3. Look at the third. 5x² has highest power 2.
   yes
4. Write the answer.
   x² + 3x and 5x²

Check. Only the expressions with an x² term are quadratic.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 + 6.
   y = 6
3. Write the answer.
   (0, 6)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 − 1.
   y = -1
3. Write the answer.
   (0, -1)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. The line of symmetry of y = x² + c goes through the turning point, at x = 0.

1. Find the turning point. The turning point of y = x² + 5 is at x = 0.
   (0, 5)
2. Write the line of symmetry. The line goes through the turning point, straight up and down.
   x = 0
3. Write the answer.
   x = 0

Check. The graph is the same on both sides of x = 0.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 − 3.
   y = -3
3. Write the answer.
   (0, -3)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. Find the value of y at x = 0 for each curve.

1. Find y for y = x² + 2. At x = 0: 0 + 2 = 2.
   2
2. Find y for y = x² − 2. At x = 0: 0 − 2 = −2.
   −2
3. Compare. The first curve is higher at x = 0.
   2 > −2
4. Write the answer.
   y = x² + 2

Check. The curves have the same shape, but one is moved up 4 units.

Method. Find the turning point of y = x² + c.

1. Use the rule. The turning point of y = x² + c is at (0, c).
   (0, 1)
2. Compare. Ali gave (1, 0), which is not the turning point.
   (1, 0) ≠ (0, 1)
3. Write the answer. Ali is not right.
   No

Check. At x = 1, y = 2, so (1, 0) is not on the curve.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 − 5.
   y = -5
3. Write the answer.
   (0, -5)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. x² is never negative, so x² + c = 0 has no solutions when c is positive.

1. Write the equation. Set y = 0.
   x² + 4 = 0
2. Rearrange. Move c to the other side.
   x² = -4
3. Decide. x² cannot be negative, so there is no real solution.
   no real solutions
4. Write the answer.
   No solutions

Check. The graph is above the x-axis, so it never crosses it.

Method. The turning point is (0, c), and a positive a means the curve opens upwards.

1. Find the turning point. With c = 1, the turning point is (0, 1).
   (0, 1)
2. Look at a. a = 2 is positive, so the curve is U-shaped (opens up).
   a > 0
3. Decide. So the turning point is the lowest point, a minimum.
   minimum
4. Write the answer.
   (0, 1), a minimum

Check. The lowest value of 2x² + 1 is 1, when x = 0.

Method. Squaring means multiplying the number by itself. A negative number times a negative number is positive.

1. Write the square. Multiply the number by itself.
   (−6) × (−6)
2. Work it out. Negative times negative gives a positive answer.
   36
3. Write the answer.
   36

Check. The square is never negative, because two negatives multiply to a positive.

Watch out. (−3)² = 9, but −3² = −9. The brackets make the difference.

Method. Without brackets, the power applies to 4 only.

1. Find 4². Four squared is 16.
   16
2. Apply the minus sign.
   −16

Check. (−4)² = 16 is different.

Method. Square each x value, multiply by 1 if needed, then add -1.

1. Find x² for x = -2. Square the number first.
   −2² = 4
2. Find x² for x = -1. Square the number first.
   −1² = 1
3. Find x² for x = 0. Square the number first.
   0² = 0
4. Work out y for each x. Multiply x² by 1 (if needed), then add -1.
   4 − 1 = 3, 1 − 1 = 0, 0 − 1 = -1
5. Write the table. Each y value goes with its x value.
   (-2, 3), (-1, 0), (0, -1), (1, 0), (2, 3)

Check. Check one value: when x = -2, y = x² − 1 gives 3.

Method. Square each x value, multiply by 2 if needed, then add 3.

1. Find x² for x = -1. Square the number first.
   −1² = 1
2. Find x² for x = 0. Square the number first.
   0² = 0
3. Find x² for x = 1. Square the number first.
   1² = 1
4. Work out y for each x. Multiply x² by 2 (if needed), then add 3.
   2 × 1 + 3 = 5, 2 × 0 + 3 = 3, 2 × 1 + 3 = 5
5. Write the table. Each y value goes with its x value.
   (-1, 5), (0, 3), (1, 5), (2, 11)

Check. Check one value: when x = -1, y = 2x² + 3 gives 5.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 25 = 0
2. Rearrange. Move c to the other side.
   x² = 25
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±5
4. Write the answer.
   x = 5 or x = −5

Check. Check: 5² = 25 and (−5)² = 25. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 + 7.
   y = 7
3. Write the answer.
   (0, 7)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. The line of symmetry of y = x² + c is x = 0.

1. Recall the rule. For y = x² + c, the graph is symmetrical about the y-axis.
   x = 0
2. Write the answer.
   x = 0

Check. The turning point (0, −2) is on this line.

Method. A quadratic has highest power 2.

1. Check each. 4x + 1 has highest power 1. x² − x has highest power 2. 2x³ has highest power 3.
   2, 1, 3
2. Write the answer.
   x² − x

Check. Only x² − x has highest power 2.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 + 5.
   y = 5
3. Write the answer.
   (0, 5)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 49 = 0
2. Rearrange. Move c to the other side.
   x² = 49
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±7
4. Write the answer.
   x = 7 or x = −7

Check. Check: 7² = 49 and (−7)² = 49. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. Square each x value, multiply by 1 if needed, then add -3.

1. Find x² for x = -2. Square the number first.
   −2² = 4
2. Find x² for x = -1. Square the number first.
   −1² = 1
3. Find x² for x = 0. Square the number first.
   0² = 0
4. Work out y for each x. Multiply x² by 1 (if needed), then add -3.
   4 − 3 = 1, 1 − 3 = -2, 0 − 3 = -3
5. Write the table. Each y value goes with its x value.
   (-2, 1), (-1, -2), (0, -3), (1, -2), (2, 1)

Check. Check one value: when x = -2, y = x² − 3 gives 1.

Method. Plot the points from the table, and join with a smooth curve.

1. Plot each point. Use the table values from the previous question.
   (−2, 1), (−1, −2), (0, −3), (1, −2), (2, 1)
2. Draw the curve. Join the points with a smooth U-shaped curve.
   smooth curve

Check. The lowest point is (0, −3).

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 36 = 0
2. Rearrange. Move c to the other side.
   x² = 36
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±6
4. Write the answer.
   x = 6 or x = −6

Check. Check: 6² = 36 and (−6)² = 36. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 + 8.
   y = 8
3. Write the answer.
   (0, 8)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. The graph of y = ax² + c is symmetrical about the y-axis.

1. Look at the form of the rule. The rule has no x term, so the curve is symmetrical about x = 0.
   x = 0
2. Write the answer.
   x = 0

Check. The turning point (0, −1) is on this line.

Method. The turning point of y = x² + c is (0, c), so c is the y value.

1. Use the turning point. The turning point is (0, 4), so the y value is c.
   c = 4
2. Write the answer.
   4

Check. y = x² + 4 has its turning point at (0, 4).

Method. Solve x² − 2 = 0 to find the roots.

1. Write the equation. Set y = 0: x² = 2.
   x² = 2
2. Take the square root. The roots are about ±1.4, not ±2.
   x ≈ ±1.4
3. Compare. Sam said ±2, which is not right.
   2 ≠ 1.4
4. Write the answer. Sam is not right.
   No

Check. 1.4² is about 2, but 2² = 4.

Method. Set y = 0 and see whether there is a solution.

1. Write the equation. Set y = 0: x² + 4 = 0.
   x² = −4
2. Decide. x² cannot be negative, so there is no solution.
   no solution
3. Write the answer. Ali is not right: the graph does not cross the x-axis.
   No

Check. The graph is above the x-axis, so it never crosses it.

Method. Set y = 0 and solve: x² = −c. Take the square root, remembering that there are two answers.

1. Write the equation. The graph crosses the x-axis where y = 0.
   x² − 81 = 0
2. Rearrange. Move c to the other side.
   x² = 81
3. Take the square root. Both a positive and a negative number square to give the same answer.
   x = ±9
4. Write the answer.
   x = 9 or x = −9

Check. Check: 9² = 81 and (−9)² = 81. The graph crosses the x-axis at both points.

Watch out. There are always two answers for x² = a positive number: one positive and one negative.

Method. The turning point of y = x² + c is at x = 0, and the graph is symmetrical about the y-axis.

1. Find the x value of the turning point. When x = 0, x² = 0, which is the smallest value of x².
   x = 0
2. Find the y value. Put x = 0 into the rule: y = 0 − 1.
   y = -1
3. Write the answer.
   (0, -1)

Check. The line of symmetry is x = 0, which is the y-axis.

Method. A larger a makes the curve narrower.

1. Compare the a values. The first has a = 1, and the second has a = 4.
   a = 4 is bigger
2. Decide. The larger a gives steeper sides, so the second is narrower.
   narrower
3. Write the answer.
   y = 4x² + 2

Check. At x = 1, the second curve is at y = 6, and the first is at y = 3.