Revision notes

Every Graphs topic condensed to one page: the facts to know, the mistakes to avoid, and the formulae. Use it as a last-minute check, or print it.

Formulae for this unit

Equation of a straight line $y = mx + c$
Gradient $\dfrac{\text{change in } y}{\text{change in } x}$
Midpoint of a line segment $\left(\dfrac{x_{1} + x_{2}}{2},\ \dfrac{y_{1} + y_{2}}{2}\right)$

Coordinates & Midpoints

Coordinates are written (x, y): along the corridor first, then up the stairs.
The midpoint of two points is the mean of the x values and the mean of the y values.
The origin is (0, 0); the x-axis is the line y = 0 and the y-axis is x = 0.
The distance between two points comes from Pythagoras on the horizontal and vertical gaps.

Watch out: Writing the y-coordinate before the x-coordinate, instead of always reading or plotting the x-coordinate first.

Straight Line Graphs

Every straight line can be written y = mx + c, where m is the gradient and c is the y-intercept.
c is where the line crosses the y-axis; m tells you how steep it is.
To plot one, make a short table of x and y values, or use the gradient and intercept.
Rearrange into y = mx + c form before reading off the gradient.

Watch out: Mixing up the gradient and y-intercept when reading y = mx + c, for example giving the y-intercept as the gradient.

Gradient, Parallel & Perpendicular Lines

Gradient is change in y divided by change in x (rise over run), read left to right.
A downhill line has a negative gradient.
Parallel lines have the same gradient.
Perpendicular lines have gradients that multiply to -1 (negative reciprocals).

Watch out: Subtracting the coordinates in the wrong order when finding the gradient, which flips the sign of the answer.

Quadratic Graphs

A quadratic graph is a parabola: a U shape when the x squared term is positive, an n shape when negative.
The roots are where the curve crosses the x-axis, found by solving y = 0.
The turning point is the vertex; completing the square gives its coordinates.
The curve is symmetrical about a vertical line through the turning point.

Watch out: Squaring a negative x value incorrectly, for example treating (-2)² as -4 instead of the correct positive value, 4.

Cubic & Reciprocal Graphs

A cubic has an x cubed term and can have up to two turning points and up to three roots.
A positive cubic runs from bottom-left to top-right; a negative one is the mirror image.
The reciprocal graph y = 1/x has two separate branches and never touches the axes.
For a reciprocal graph the axes are asymptotes: the curve gets close but never reaches them.

Watch out: Making an arithmetic error when cubing a negative number, for example treating (-2)³ as 8 instead of the correct value, -8.

Real-Life Graphs

On a distance-time graph the gradient is the speed, and a horizontal line means stopped.
On a speed-time graph the gradient is the acceleration and the area is the distance travelled.
A steeper line means a faster rate of change.
Read the axis labels and units before you interpret anything.

Watch out: Reading a conversion graph in the wrong direction, for example starting from the wrong axis when converting a value.

Graphical Solutions to Equations

The solution to two graphs is where they cross: read off the x and y coordinates.
To solve f(x) = k, draw the line y = k and find where it meets the curve.
A curve and a line can cross twice, once, or not at all.
Rearrange the equation so one side matches a graph you have already drawn.

Watch out: Giving only one intersection point for a quadratic equation, when a curve and a line usually cross at two points, giving two solutions.

Graph Transformations

y = f(x) + a moves the graph up by a; y = f(x) - a moves it down.
y = f(x + a) moves the graph left by a; y = f(x - a) moves it right (inside the bracket does the opposite of what you expect).
y = -f(x) reflects the graph in the x-axis; y = f(-x) reflects it in the y-axis.
y = a f(x) stretches the graph vertically by scale factor a.

Watch out: Getting the direction of a horizontal translation backwards, for example thinking y = f(x - 2) shifts the graph to the left instead of to the right.

Area Under a Graph & Rate of Change

The area under a speed-time graph is the distance travelled.
Split the area into triangles, rectangles and trapeziums, then add them up.
The trapezium rule estimates the area under a curve using vertical strips.
Straight-line sections give an exact area; curved sections give an estimate.

Watch out: Forgetting to multiply by h ÷ 2 (half the strip width) when applying the trapezium rule.

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