GCSE Maths: Recognising graphs from their equations (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)
Recognising graphs from their equations
There are four families of graph in this chapter and in Chapter 12. You should be able to recognise each one from its equation and from its shape, and to sketch it. A sketch is a quick drawing that shows the correct shape and any important points, such as where it crosses the axes. It does not need a table of values or an accurate scale.
To name a graph from its equation, look at the highest power of x, or whether x is on the bottom of a fraction:
- Linear (highest power x, for example y = 2x + 1): a straight line. The number in front of x is the gradient: positive goes up to the right, negative goes down.
- Quadratic (highest power x²): a parabola. Positive x² gives a U, negative gives an n.
- Cubic (highest power x³): an S-shaped curve. Positive x³ goes from bottom-left to top-right, negative goes from top-left to bottom-right.
- Reciprocal (x on the bottom, for example y = 5 ÷ x): two branches. Positive number on top: top-right and bottom-left. Negative: top-left and bottom-right.
Once you know the family, use any numbers added or subtracted to find the y-intercept: put x = 0 into the rule. For y = x² − 3 the intercept is −3, so the U-shaped curve crosses the y-axis at (0, −3).
Key ideas
- Highest power x: straight line. Highest power x²: parabola (U if positive, n if negative).
- Highest power x³: cubic S-shape (up if positive, down if negative).
- x on the bottom of a fraction: reciprocal, two branches, never touching the axes.
- The y-intercept comes from putting x = 0 into the rule.
- A sketch shows the shape and the key points (intercepts, turning point) without an accurate scale.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x to name the graph, then use the sign of that term and the constant term.
- Find the highest power of x. In x³ − 2 the highest power is 3.Highest power: x³
- Name and describe the graph. A graph with x³ is a cubic. The number in front of x³ is +1 (positive), so it goes from bottom-left to top-right.Cubic, rising from bottom-left to top-right
- Find the y-intercept. On the y-axis x = 0. Put x = 0 into the rule.y = 0³ − 2 = −2, so it crosses at (0, −2)
Check. The point (0, −2) fits the rule, and a cubic does cross the y-axis once. ✓
Watch out. Do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name the graph from each equation, then find the matching picture.
- Name each graph. Look at the highest power of x, or x on the bottom.1: 2x + 1 is linear.
2: x² + 1 is quadratic.
3: x³ is cubic.
4: 4 ÷ x is reciprocal. - Match the shapes. Linear is a straight line (B). Quadratic is a U (D). Cubic is an S (A). Reciprocal has two branches (C).A: S-shape, B: straight line, C: two branches, D: U-shape
- Write the matches. 1 → B, 2 → D, 3 → A, 4 → C
Check. Check one: y = x² + 1 is always at least 1, so it never goes below the x-axis: the U in D. ✓
Watch out. Name the graph first.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Use the shape to decide the family and the direction, then the y-intercept to decide between the rest.
- Describe the shape. The curve is an n-shape (opening downwards) with its highest point on the y-axis.n-shaped parabola, so a quadratic with a negative x² term
- Remove the wrong families. C is a cubic, D is a reciprocal and A has a positive x² term (a U-shape).Only B has a negative x² term: 3 − x² = −x² + 3
- Check the y-intercept. Put x = 0 into B.y = 3 − 0 = 3, which matches the graph
- Write the answer. B: y = 3 − x²
Check. Test the root: x² = 3 gives x = ±1.7 (about), where the curve meets the x-axis in the picture. ✓
Watch out. Check the sign of the x² term, not just the number 3.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name the family, then decide its direction.
- Name the graph. The highest power is x³, so it is a cubic.Cubic
- Find the direction. The number in front of x³ is negative (−1), so the graph goes from top-left to bottom-right.Falls from top-left to bottom-right
- Find a key point. Put x = 0 in the rule.y = 0, so it passes through the origin (0, 0)
- Draw. Draw smooth axes, then a smooth S-shaped curve that falls through the origin.See the sketch below.
Check. At x = 1, y = −1 (below the axis) and at x = −1, y = 1 (above): consistent with falling. ✓
Watch out. Label the origin and keep the curve smooth.
Watch out
A graph is quadratic only if the highest power is x². Do not mix up the shapes: x³ is S-shaped, not U-shaped. And a reciprocal graph never touches either axis. Always check the sign: a minus in front of the power turns the picture upside down (or flips it).Exam tip
Decide the family first (highest power), then the sign, then use the y-intercept (put x = 0) to choose between similar graphs.Exercise 13E Recognising and sketching graphs
Non-calculatorKey words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is just x (power 1), the graph is a straight line.y = 3x − 2: the highest power is x.
- Use the number in front of x (the gradient). A positive gradient slopes up as you move right; a negative gradient slopes down.The gradient is 3, which is positive.
- Describe the shape. Put the two facts together.Linear: straight line going up as you move right.
Check. Test the idea with a point: when x = 1, y = 1, and when x = −1, y = −5. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x², the graph is a quadratic (a parabola).y = x2 + 4: the highest power is x².
- Use the sign of the x² term. A positive x² term gives a U-shape. A negative one gives an n-shape.The number in front of x² is 1, which is positive.
- Describe the shape. Put the two facts together.Quadratic: U-shaped parabola (has a lowest point).
Check. Test the idea with a point: when x = 1, y = 5, and when x = −1, y = 5. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x³, the graph is a cubic.y = x3 − 1: the highest power is x³.
- Use the sign of the x³ term. A positive x³ term makes the graph go from bottom-left to top-right. A negative one makes it go from top-left to bottom-right.The number in front of x³ is 1, which is positive.
- Describe the shape. Put the two facts together.Cubic: cubic curve rising from bottom-left to top-right.
Check. Test the idea with a point: when x = 1, y = 0, and when x = −1, y = −2. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look at where x is. In this rule x is on the bottom of a fraction. A rule of the form number ÷ x is a reciprocal graph.y = 5x
- Use the sign of the number on top. A positive number on top gives branches in the top-right and bottom-left. A negative number gives branches in the top-left and bottom-right.The number on top is 5, which is positive.
- Describe the shape. Put the two facts together.Reciprocal: reciprocal curve in two branches, in the top-right and bottom-left parts of the grid.
Check. Test the idea with a point: when x = 1, y = 5, and when x = −1, y = −5. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x², the graph is a quadratic (a parabola).y = −2x2: the highest power is x².
- Use the sign of the x² term. A positive x² term gives a U-shape. A negative one gives an n-shape.The number in front of x² is −2, which is negative.
- Describe the shape. Put the two facts together.Quadratic: n-shaped parabola (has a highest point).
Check. Test the idea with a point: when x = 1, y = −2, and when x = −1, y = −2. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look at where x is. In this rule x is on the bottom of a fraction. A rule of the form number ÷ x is a reciprocal graph.y = −3x
- Use the sign of the number on top. A positive number on top gives branches in the top-right and bottom-left. A negative number gives branches in the top-left and bottom-right.The number on top is −3, which is negative.
- Describe the shape. Put the two facts together.Reciprocal: reciprocal curve in two branches, in the top-left and bottom-right parts of the grid.
Check. Test the idea with a point: when x = 1, y = −3, and when x = −1, y = 3. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name each graph from its equation, then find the matching picture.
- Say what kind of graph the equation gives. Look at the highest power of x (or x on the bottom of a fraction).y = 2x + 1 is a linear graph: straight line going up as you move right.
- Look for that shape among the graphs. Only graph B has this shape.Graph B is a straight line going up as you move right.
- Write the match. y = 2x + 1 matches graph B
Check. Test a point: when x = 1 the rule gives y = 3, which fits the picture. ✓
Watch out. Match by shape first, then use signs to choose between similar shapes.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name each graph from its equation, then find the matching picture.
- Say what kind of graph the equation gives. Look at the highest power of x (or x on the bottom of a fraction).y = x2 + 1 is a quadratic graph: U-shaped parabola (has a lowest point).
- Look for that shape among the graphs. Only graph D has this shape.Graph D is a U-shaped parabola (has a lowest point).
- Write the match. y = x2 + 1 matches graph D
Check. Test a point: when x = 1 the rule gives y = 2, which fits the picture. ✓
Watch out. Match by shape first, then use signs to choose between similar shapes.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name each graph from its equation, then find the matching picture.
- Say what kind of graph the equation gives. Look at the highest power of x (or x on the bottom of a fraction).y = x3 is a cubic graph: cubic curve rising from bottom-left to top-right.
- Look for that shape among the graphs. Only graph A has this shape.Graph A is a cubic curve rising from bottom-left to top-right.
- Write the match. y = x3 matches graph A
Check. Test a point: when x = 1 the rule gives y = 1, which fits the picture. ✓
Watch out. Match by shape first, then use signs to choose between similar shapes.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name each graph from its equation, then find the matching picture.
- Say what kind of graph the equation gives. Look at the highest power of x (or x on the bottom of a fraction).y = 4x is a reciprocal graph: reciprocal curve in two branches, in the top-right and bottom-left parts of the grid.
- Look for that shape among the graphs. Only graph C has this shape.Graph C is a reciprocal curve in two branches, in the top-right and bottom-left parts of the grid.
- Write the match. y = 4x matches graph C
Check. Test a point: when x = 1 the rule gives y = 4, which fits the picture. ✓
Watch out. Match by shape first, then use signs to choose between similar shapes.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name each graph from its equation, then find the matching picture.
- Say what kind of graph the equation gives. Look at the highest power of x (or x on the bottom of a fraction).y = −x + 3 is a linear graph: straight line going down as you move right.
- Look for that shape among the graphs. Only graph C has this shape.Graph C is a straight line going down as you move right.
- Write the match. y = −x + 3 matches graph C
Check. Test a point: when x = 1 the rule gives y = 2, which fits the picture. ✓
Watch out. Match by shape first, then use signs to choose between similar shapes.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name each graph from its equation, then find the matching picture.
- Say what kind of graph the equation gives. Look at the highest power of x (or x on the bottom of a fraction).y = −x2 is a quadratic graph: n-shaped parabola (has a highest point).
- Look for that shape among the graphs. Only graph B has this shape.Graph B is a n-shaped parabola (has a highest point).
- Write the match. y = −x2 matches graph B
Check. Test a point: when x = 1 the rule gives y = −1, which fits the picture. ✓
Watch out. Match by shape first, then use signs to choose between similar shapes.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name each graph from its equation, then find the matching picture.
- Say what kind of graph the equation gives. Look at the highest power of x (or x on the bottom of a fraction).y = −x3 is a cubic graph: cubic curve falling from top-left to bottom-right.
- Look for that shape among the graphs. Only graph D has this shape.Graph D is a cubic curve falling from top-left to bottom-right.
- Write the match. y = −x3 matches graph D
Check. Test a point: when x = 1 the rule gives y = −1, which fits the picture. ✓
Watch out. Match by shape first, then use signs to choose between similar shapes.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Name each graph from its equation, then find the matching picture.
- Say what kind of graph the equation gives. Look at the highest power of x (or x on the bottom of a fraction).y = −2x is a reciprocal graph: reciprocal curve in two branches, in the top-left and bottom-right parts of the grid.
- Look for that shape among the graphs. Only graph A has this shape.Graph A is a reciprocal curve in two branches, in the top-left and bottom-right parts of the grid.
- Write the match. y = −2x matches graph A
Check. Test a point: when x = 1 the rule gives y = −2, which fits the picture. ✓
Watch out. Match by shape first, then use signs to choose between similar shapes.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x², the graph is a quadratic (a parabola).y = x2: the highest power is x².
- Use the sign of the x² term. A positive x² term gives a U-shape. A negative one gives an n-shape.The number in front of x² is 1, which is positive.
- Describe the shape. Put the two facts together.Quadratic: U-shaped parabola (has a lowest point).
Check. Test the idea with a point: when x = 1, y = 1, and when x = −1, y = 1. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x², the graph is a quadratic (a parabola).y = −x2: the highest power is x².
- Use the sign of the x² term. A positive x² term gives a U-shape. A negative one gives an n-shape.The number in front of x² is −1, which is negative.
- Describe the shape. Put the two facts together.Quadratic: n-shaped parabola (has a highest point).
Check. Test the idea with a point: when x = 1, y = −1, and when x = −1, y = −1. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x³, the graph is a cubic.y = x3: the highest power is x³.
- Use the sign of the x³ term. A positive x³ term makes the graph go from bottom-left to top-right. A negative one makes it go from top-left to bottom-right.The number in front of x³ is 1, which is positive.
- Describe the shape. Put the two facts together.Cubic: cubic curve rising from bottom-left to top-right.
Check. Test the idea with a point: when x = 1, y = 1, and when x = −1, y = −1. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look for the highest power of x. If the highest power is x³, the graph is a cubic.y = −x3: the highest power is x³.
- Use the sign of the x³ term. A positive x³ term makes the graph go from bottom-left to top-right. A negative one makes it go from top-left to bottom-right.The number in front of x³ is −1, which is negative.
- Describe the shape. Put the two facts together.Cubic: cubic curve falling from top-left to bottom-right.
Check. Test the idea with a point: when x = 1, y = −1, and when x = −1, y = 1. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look at where x is. In this rule x is on the bottom of a fraction. A rule of the form number ÷ x is a reciprocal graph.y = 2x
- Use the sign of the number on top. A positive number on top gives branches in the top-right and bottom-left. A negative number gives branches in the top-left and bottom-right.The number on top is 2, which is positive.
- Describe the shape. Put the two facts together.Reciprocal: reciprocal curve in two branches, in the top-right and bottom-left parts of the grid.
Check. Test the idea with a point: when x = 1, y = 2, and when x = −1, y = −2. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Look at the highest power of x (or whether x is on the bottom of a fraction) to name the graph, then use the sign to describe its direction.
- Look at where x is. In this rule x is on the bottom of a fraction. A rule of the form number ÷ x is a reciprocal graph.y = −2x
- Use the sign of the number on top. A positive number on top gives branches in the top-right and bottom-left. A negative number gives branches in the top-left and bottom-right.The number on top is −2, which is negative.
- Describe the shape. Put the two facts together.Reciprocal: reciprocal curve in two branches, in the top-left and bottom-right parts of the grid.
Check. Test the idea with a point: when x = 1, y = −2, and when x = −1, y = 2. These agree with the shape described. ✓
Watch out. A graph with x² is a curve with one turning point; do not call a cubic a quadratic.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Use the shape and the y-intercept.
- Find the shape. x² has a positive number in front, so the graph is U-shaped. This rules out C.U-shaped
- Find the y-intercept. Put x = 0 in the rule.y = 0 − 3 = −3, so the intercept is (0, −3). This rules out A.
- Choose. Only B is U-shaped with intercept −3.B
Check. B passes through (0, −3), and (3, 6) is also on the curve: 9 − 3 = 6. ✓
Watch out. Use both the shape and the intercept.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Test the statement using what you know about the shape or by substituting.
- Decide. The x² term can be negative, giving an n-shape.
- Write the answer. Use the reason to decide.False
Check. y = −x² is n-shaped.
Watch out. Always give a reason.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Test the statement using what you know about the shape or by substituting.
- Decide. The highest power of x is 1, so it is a straight line.
- Write the answer. Use the reason to decide.False
Check. Straight line with gradient 2.
Watch out. Always give a reason.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Test the statement using what you know about the shape or by substituting.
- Decide. (−1) × (−1) × (−1) = −1.
- Write the answer. Use the reason to decide.True
Check. The point fits the rule.
Watch out. Always give a reason.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Test the statement using what you know about the shape or by substituting.
- Decide. You cannot divide by 0.
- Write the answer. Use the reason to decide.False
Check. There is no point with x = 0.
Watch out. Always give a reason.
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. A turning point is where a graph changes from going down to going up (or the reverse). Think about each shape.
- Straight line. A line always goes the same way.No turning point
- Quadratic. A U or n shape turns once at its lowest or highest point.Has a turning point
- y = x³. It rises all the way from bottom-left to top-right and never turns.No turning point
- Reciprocal. Each branch keeps going the same way.No turning point
- Answer. y = x² − 3
Check. The table for x² − 3 goes down then up, but the tables for the others never change direction. ✓
Watch out. A cubic like y = x³ − 4x does have turns; the simple y = x³ does not.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Substitute the x-value into the rule and compare with the y-value.
- Substitute. Cube the x-value. Use brackets for negatives.x = 2: y = 23 = 8
- Compare. The point has y = 8.8 = 8, so the point is on the graph
Check. A point on y = x³ always has y of the same sign as x. ✓
Watch out. Only an exact match counts.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Substitute the x-value into the rule and compare with the y-value.
- Substitute. Cube the x-value. Use brackets for negatives.x = −2: y = (−2)3 = −8
- Compare. The point has y = −8.−8 = −8, so the point is on the graph
Check. A point on y = x³ always has y of the same sign as x. ✓
Watch out. Only an exact match counts.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Substitute the x-value into the rule and compare with the y-value.
- Substitute. Cube the x-value. Use brackets for negatives.x = 1: y = 13 = 1
- Compare. The point has y = 2.1 ≠ 2, so the point is not on the graph
Check. A point on y = x³ always has y of the same sign as x. ✓
Watch out. Only an exact match counts.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Substitute the x-value into the rule and compare with the y-value.
- Substitute. Cube the x-value. Use brackets for negatives.x = −1: y = (−1)3 = −1
- Compare. The point has y = 1.−1 ≠ 1, so the point is not on the graph
Check. A point on y = x³ always has y of the same sign as x. ✓
Watch out. Only an exact match counts.
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. Use the shape, the y-intercept and the roots to find the equation.
- Use the shape. An n-shape needs a negative x² term. This leaves 4 − x².Only y = 4 − x² has a negative x²
- Check the y-intercept. Put x = 0.y = 4 − 0 = 4 ✓
- Check the roots. Put y = 0: 4 − x² = 0 gives x² = 4.x = 2 or x = −2 ✓
Check. All three facts match. ✓
Watch out. Test every fact in the question.
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 origin is (0, 0). Substitute x = 0 into the rule and see whether y is 0.
- Substitute x = 0. The origin has x = 0.y = 0² + 1 = 0 + 1 = 1
- Compare. The origin has y = 0, but the rule gives y = 1.1 ≠ 0
- Conclude. At x = 0 the graph is at (0, 1), above the origin.The graph does not pass through (0, 0). It crosses the y-axis at (0, 1).
Check. x² is never negative, so x² + 1 is never less than 1: it cannot be 0. ✓
Watch out. Show the substitution; do not just say "no".
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Compare the two rules at x = 2 and x = −2.
- Work out y = x² at 2 and −2. Squares of opposite numbers are the same.2² = 4 and (−2)² = 4
- Work out y = x³ at 2 and −2. Cubes of opposite numbers are opposites.2³ = 8 and (−2)³ = −8
- Explain. For y = x² the left and right sides match, so it is symmetrical about the y-axis (a U-shape). For y = x³ the y-value changes sign, so one end goes up and the other goes down (an S-shape).x²: same y, U-shape. x³: opposite y, S-shape.
Check. Try x = 3: 9 and 9 for x²; 27 and −27 for x³. ✓
Watch out. Compare the values for x and −x.
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. Use the roots to choose the brackets, then the y-intercept to decide the sign.
- Find the brackets from the roots. A root at x = 1 gives the bracket (x − 1). A root at x = −3 gives (x + 3).Brackets (x − 1)(x + 3): choices 1 and 2
- Find the y-intercept of choice 1. Put x = 0: (0 − 1)(0 + 3) = (−1)(3).y = −3 (too low: we need +3)
- Find the y-intercept of choice 2. The extra minus changes the sign.y = −(−1)(3) = 3 ✓
- Write the answer. This is also n-shaped, as it has −x².y = −(x − 1)(x + 3)
Check. Expand: −(x² + 2x − 3) = −x² − 2x + 3, which has y-intercept 3. ✓
Watch out. Roots give brackets with the sign flipped.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. Factorise to find the roots, then use the sign of x³ for the direction.
- Find the roots. Put y = 0 and factorise: take out x, then use the difference of two squares.x³ − 4x = x(x² − 4) = x(x − 2)(x + 2) = 0, so x = 0, 2 or −2
- Find the direction. The x³ term is positive, so the right-hand end goes up and the left-hand end goes down.Bottom-left to top-right
- Sketch. Draw a smooth curve through the three roots with a hump between −2 and 0 and a dip between 0 and 2.See the sketch in the answer.
Check. At x = 1: y = 1 − 4 = −3, below the axis, between the roots 0 and 2: the dip. ✓
Watch out. Label all three crossing points.
Key words. A cubic is a rule where the highest power of x is x³ (x cubed). A reciprocal is 1 divided by a number, so the reciprocal of 4 is 1 ÷ 4 = 0.25. An asymptote is a line that a curve gets closer and closer to but never touches.
Method. The rule is the negative of part (a), so the graph is part (a) flipped over.
- Find the roots. Put y = 0: 4x − x³ = x(4 − x²) = x(2 − x)(2 + x).x = 0, 2 or −2
- Find the direction. The x³ term is negative, so the right-hand end goes down and the left-hand end goes up.Top-left to bottom-right
- Sketch. Same roots as part (a), but a dip between −2 and 0 and a hump between 0 and 2... reversed.See the sketch in the answer.
Check. At x = 1: y = 4 − 1 = 3, above the axis between 0 and 2: the hump. ✓
Watch out. A minus in front flips the graph over the x-axis.
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. Substitute a general x and then −x and compare.
- Substitute x. The value of y is a times x squared, plus c.y = a x² + c
- Substitute −x. Squaring a negative gives the same as squaring the positive: (−x)² = x².y = a (−x)² + c = a x² + c
- Compare. The two y-values are equal for every x, so every point (x, y) has a mirror image (−x, y).The graph is symmetrical about the y-axis.
Check. Test: y = 3x² + 1 at x = 2 and −2: 13 and 13. ✓
Watch out. This works because (−x)² = x². It would fail with an x term, e.g. x² + x.