What you need to know

We’ve seen how to draw straight lines (https://mathsmadeeasy.co.uk/gcse-maths-revision/drawing-straight-line-graphs-gcse-maths-revision-worksheets/), so now we’re going to look at the weirder, curvier, more interesting kinds of graphs. However, the way that we plot them will be the same: by using an equation to fill a table of coordinates, plotting these coordinates, and joining the points together – only this time, the lines will be curved. The first type of graph we will see is a quadratic.

Example: Using the equation y=x^2-x-5, draw a table of coordinates from x=-2 to x=3. Use the values to plot the graph between these values of x.

We draw this table by substituting the x values into the equation. For example, when x=-2, we get

 

y=(-2)^2-(-2)-5=4+2-5=1.

Carrying this on with the other numbers and we get a filled table that looks like the one above. Then, plotting these points on a pair of axes and joining them up with a curve, we get the graph to the right.

 

The shape of a quadratic is known as a parabola.

 

The only very noticeable way it can be different is when the x^2 term in the quadratic has a negative sign in front of it, in which case the parabola is flipped upside-down.

The next type of graph we will see is a reciprocal graph.

 

Example: Using the equation y=\frac{1}{x}, draw a table of coordinates from x=1 to x=5. Use the values to plot the graph between x=0 and x=5.

 

We draw this table by substituting the x values into the equation. For example, for x=2, we get

 

y=\frac{1}{2}=0.5.

 

The next type of graph we will see is a cubic.

 

Example: Using the equation y=x^3-2x^2, draw a table of coordinates from x=-1 to x=3. Use the values to plot the graph between these x values.

 

We draw this table by substituting the x values into the equation. For example, for x=1, we get

 

y=1^3-2(1)^2=-1.

Carrying this on with the other numbers and we get a filled table like the one above. Then, notice that the question asks us to draw from x=0 but we’ve only calculated from x= 1. This is because at x=0 this graph has asymptotes – line which the graph will get closer and closer to but never meet – along both axes.

 

We *technically* can’t do what the question asks, because we can’t draw up to infinity, but we can plot the points, join them up, and show that as the graph goes closer to 0 on the x -axis, its y value gets bigger and bigger. Question: what does this graph look like for negative values of x?

The next graph we will see is an exponential graph and is only for those studying the higher course.

 

Example: Using the equation y=3^x, draw a table of coordinates from x=-3 to x=2. Use the values to plot the graph between these x values.

 

We draw this table by substituting the x values into the equation. For example, for x=1, we get

 

y=3^1=3.

Carrying this on with the rest of the numbers, we get the table above. Then, plotting these points and joining them with a curve, we get the graph to the right.

 

The exponential graph also has an asymptote along the x-axis. Its shape varies very little, except that when the base of the exponential (here, the function is 3^x so the base is 3) is a number between 0 and 1, the shape of the graph is a mirror image of this one. Specifically, a reflection in the y-axis.

Example Questions

1) Using the equation y=x^2+4x-9, complete the table of coordinates below. Use these coordinates to plot the graph between x=-5 and x=2.

 

 

Answer

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=2,

 

y=(-4)^2+4(-4)-9=16-16-9=-9

 

Continuing this with the rest of the x values, we get the completed table below.

 

 

Then, plotting these coordinates on a pair of axes and joining them with a curve, we get the graph below.

 

 

 

2) Using the equation y=x^3+3x^2-4, complete the table of coordinates below. Use these coordinates to plot the graph between x=-4 and x=1.

 

 

Answer

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=-2,

 

y=(-2)^3+3(-2)^2-4=-8+12-4=0

 

Continuing this with the rest of the x values, we get the completed table below.

 

 

Then, plotting these points on a pair of axes and joining them with a curve, we get the graph below.

 

 

 

3) (HIGHER ONLY) Using the equation y=(0.2)^x, complete the table of coordinates below. Use these coordinates to plot the graph between x=-1 and x=4.

 

 

Answer

We will complete this table by substituting in the values of x to get the missing values of y. For example, when x=2,

 

y=(0.2)^2=0.04

 

Continuing this with the rest of the x values, we get the completed table below.

 

 

Then, plotting these points on a pair of axes (to the best of your ability – when some of them are so small they’re naturally going to practically end up on the x-axis) and joining them with a curve, we get the graph below.

 

 

 

From Maths tutors in Leeds to GCSE maths teachers in London, the quadratic graph and harder graphs to learn resources on this page make excellent learning aids and can also be used to set homework. As a private tutor or classroom teacher you will be always on the look out for resources to help your pupils as much as possible and the quadratic graph questions and harder graphs worksheets on this page will do just that.