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 $x=2$.

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 $x=1$.

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 $x=4$.

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.