Graphs and Functions

Every equation draws a shape

A graph is a picture of an equation. Every point on the line is a pair of numbers that makes the equation true, so the whole shape is just every solution drawn at once. The useful part is that the KIND of equation decides the SHAPE, and you can learn to tell them apart at a glance. - A linear equation, like y = 2x plus 1, with no squared or higher term, draws a straight line. - A quadratic, with an x squared term, draws a parabola, a smooth symmetrical curve shaped like a U. - A cubic, with an x cubed term, draws an S-shaped curve. - A reciprocal, like y = 1 divided by x, draws two separate curves in opposite corners. The other big idea is the function: a rule that takes an input and gives back one output. This module covers both, and how to plot a graph and read one.

Words for graphs

Five terms you need before you use them. Learn what each one means.

Match the equation to its shape

Straight line against parabola

The two graphs you meet most often are the straight line and the parabola. Knowing what to look for tells them apart instantly.

Spot the true graph facts

Tap the TWO statements that are true.

• A linear graph is a straight line
• A quadratic graph is a curve called a parabola
• A cubic graph is a straight line
• A reciprocal graph is one straight line through the middle

Use the function

A function is given by f of x = 3x. This means multiply the input by 3. What is f of 2?

• 6
• 5
• 32
• 9

How to plot a graph

Plotting any graph from its equation follows the same routine, and it never fails. First, choose a few values of x, for example minus 2, minus 1, 0, 1 and 2. Then work out y for each one by putting the x value into the equation. Set the pairs out in a small table so you do not lose track. Then plot each pair as a point and join them: with a ruler for a straight line, and with a smooth freehand curve for anything with a squared or cubed term. ⚠️ One misplaced point is obvious once the others line up, so plotting several is its own check.

Evaluate the function

A function is given by f of x = 2x plus 1. To find f of 3, replace x with 3, then work out 2 times 3 plus 1.

Plot the parabola

The graph of y = x squared is a parabola. Find y for each x by squaring it, then plot the points: at x = -2 plot 4, at x = -1 plot 1, at x = 0 plot 0, at x = 1 plot 1, and at x = 2 plot 4.

Order the plotting steps

Put the steps for plotting a graph from its equation in order.

• Choose a few values of x
• Work out the y value for each one from the equation
• Plot each pair as a point
• Join the points with a line or a smooth curve

Build the graph rule

Choose the words that complete the rule for a straight-line graph.

Complete the graph facts

A straight-line graph has the equation y = mx plus c, where m is the _____ and c tells you where the line crosses the _____. A quadratic graph is a curve called a _____. Working out the output of a function for a given input is called _____ the function.

Reading a straight line

Here is a worked example so the method is clear. A straight line has the equation y = 2x plus 1. The gradient is 2. That is the number in front of x, so for every one step to the right the line goes two steps up. A gradient of 2 is fairly steep and slopes upward. The y-intercept is 1. That is the number on its own, so the line crosses the y-axis at a height of 1. To find a point, choose an x. Put x = 3 into the equation: 2 times 3 plus 1 is 7, so the line passes through the point 3 comma 7. Choosing two or three x values and joining the points is all it takes to draw the whole line.

Name and use the graph

Read each case and choose the best answer, then think about why.

• You are given y = 3x minus 2. Without plotting it, what shape is its graph and why?
• For the function f of x = 4x, what is f of 5?
• A graph is a smooth U-shaped curve, symmetrical, with one lowest point. What kind of graph is it?

Explain graphs and functions

A student in the year below cannot tell the graph types apart or use function notation. Explain graphs and functions using the ideas from this module.

• Explain what the graph of a linear equation looks like, and name the gradient and the y-intercept
• Explain what a quadratic graph looks like, including its turning point
• Name one other type of graph and describe its shape
• Explain what function notation such as f of x means, with a worked example of evaluating a function
• Finish with the steps you would follow to plot a graph from its equation