Shapes of graphs Revision Notes for OCR GCSE Maths

Shapes of Graphs

Understanding the Shapes of Graphs

When dealing with quadratic and cubic graphs, it's important to recognise the general shapes based on the type of equation you are working with. This can help you predict the graph's behaviour and avoid errors when plotting.

1. Positive $x$

Equation: The highest power of $x$ is $1$ , and the $x$ term is positive.
General Shape: A straight line that slopes upwards from left to right.

infoNote

Examples:

y=2x+3

y=x−8

y=5x

y=9x−6

Graph Explanation:

These graphs will always rise as you move from left to right on the coordinate plane.
The slope (gradient) is positive, indicating that as $x$ increases, $y$ also increases.

2. Negative $x$

Equation: The highest power of $x$ is $1$ , and the $x$ term is negative.
General Shape: A straight line that slopes downwards from left to right.

infoNote

Examples:

y=−5x+3

y=−x−3

y=−7x

y=5−6x

Graph Explanation:

These graphs will always fall as you move from left to right on the coordinate plane.
The slope (gradient) is negative, indicating that as $x$ increases, $y$ decreases.

3. Positive $x^2$

Equation: The highest power of $x$ is $2$ , and the $x^2$ term is positive.
General Shape: A parabola that opens upwards.

infoNote

Examples:

y=x^2

y=x^2+5

y=x^2−3x+2

y=3x^2+2x−6

Explanation:

These equations produce U-shaped curves (parabolas) that open upwards.
The graph indicates that as x moves away from $0, y$ increases.

4. Negative $x^2$

Equation: The highest power of $x$ is $2$ , and the $x^2$ term is negative.
General Shape: A parabola that opens downwards.

infoNote

Examples:

y=−x^2

y=−2x^2+4

y=−2(x^2−5x+5)

y=5+3x−x^2

Explanation:

These equations produce an inverted U-shaped curve (parabola) that opens downwards.
The graph shows that as $x$ moves away from $0, y$ decreases.

5. Positive $x^3$

Equation: The highest power of $x$ is $3$ , and the $x^3$ term is positive.
General Shape: A cubic curve that starts from the bottom left, curves upwards through the origin, and then continues to rise steeply.

infoNote

Examples:

y=x^3

y=x^3+10

y=x^3−2x^2−4x+2

y=2x^3−6x

Explanation:

These equations produce an S-shaped curve that starts low, curves upward through the origin (or another point if shifted), and rises sharply as $x$ increases.

6. Negative $x^3$

Equation: The highest power of $x$ is $3$ , and the $x^3$ term is negative.
General Shape: A cubic curve that starts from the top left, curves downwards through the origin, and then continues to fall steeply.

infoNote

Examples:

y=−x^3

y=−5x^3+2

y=−4x^3+x^2+5

y=5+3x+5x^2−x^3

Explanation:

These equations produce an inverted S-shaped curve that starts high, curves downward through the origin (or another point if shifted), and falls sharply as $x$ increases.

7. Positive Reciprocal

Equation: Contains a fraction with a positive $x$ on the bottom.
General Shape: A hyperbola that exists in the first and third quadrants of the Cartesian plane. The graph approaches but never touches the $x-$ axis or $y-$ axis (asymptotes).

infoNote

Examples:

y=\frac1x

y=\frac5x

y=\frac{7}{2x}

y=\frac{3}{4x}+2

Explanation:

These equations produce a graph that falls as $x$ increases in the first quadrant and rises as $x$ decreases in the third quadrant. As $x$ approaches zero, the function's value increases or decreases sharply, indicating that the graph will never touch the axes (asymptotes).

8. Negative Reciprocal

Equation: Contains a fraction with a negative $x$ on the bottom.
General Shape: A hyperbola that exists in the second and fourth quadrants of the Cartesian plane. Like the positive reciprocal, the graph approaches but never touches the $x-$ axis or $y-$ axis (asymptotes).

infoNote

Examples:

y=−\frac1x

y=−\frac5x

y=−\frac{7}{2x}

y=2−\frac2x

Explanation:

These equations produce a graph that rises as $x$ decreases in the second quadrant and falls as $x$ increases in the fourth quadrant. Again, as $x$ approaches zero, the graph's value increases or decreases sharply, ensuring the graph does not touch the axes (asymptotes).

Shapes of graphs (OCR GCSE Maths): Revision Notes

Shapes of Graphs

Understanding the Shapes of Graphs

1. Positive xxx

2. Negative xxx

3. Positive x2x^2x2

4. Negative x2x^2x2

5. Positive x3x^3x3

6. Negative x3x^3x3

7. Positive Reciprocal

8. Negative Reciprocal

Explore OCR GCSE Maths Model Answers by Topics

Rules of algebra

Single brackets

Factorising

Solving equations

Double brackets

More factorising - quadratics

Solving quadratic equations

Simultaneous equations

Inequalities

Indices

Straight line graphs

Quadratic and cubic graphs

Shapes of graphs

Travel graphs and story graphs

Surds

Explore OCR GCSE Maths Quizzes by Topics

Rules of algebra

Single brackets

Factorising

Solving equations

Double brackets

More factorising - quadratics

Solving quadratic equations

Simultaneous equations

Inequalities

Indices

Straight line graphs

Quadratic and cubic graphs

Shapes of graphs

Travel graphs and story graphs

Surds

Explore OCR GCSE Maths Flashcards by Topics

Rules of algebra

Single brackets

Factorising

Solving equations

Double brackets

More factorising - quadratics

Solving quadratic equations

Simultaneous equations

Inequalities

Indices

Straight line graphs

Quadratic and cubic graphs

Shapes of graphs

Travel graphs and story graphs

Surds

Join 100,000+ GCSE students studying Revision Notes with us.

1. Positive $x$

2. Negative $x$

3. Positive $x^2$

4. Negative $x^2$

5. Positive $x^3$

6. Negative $x^3$