Grade 12

Mixed Graphs I (linear, quad, exp, circles)

Lesson

We've looked at heaps of different graphs that we can plot on a number line. In this chapter, we are going to look at how to apply what we've learnt. We'll start by recapping some features of different types of graphs. Make sure you're familiar with how to find all these different features.

Linear Equations (Straight Lines)

The general form of a linear equation is $ax+by+c=0$ax+by+c=0. You may be asked to calculate the:

Midpoint
Slope
Distance
$x$x and $y$y intercepts
Equation of a straight line

Graphing Parabolas

The general equation of a parabola is $ax^2+b^y+c=0$ax2+by+c=0. You may be asked to find the:

Equation of the axis of symmetry
Vertex
Minimum/ maximum value
$x$x and $y$y intercepts
Equation of the parabola

Graphing Circles

The general equation of a circle is:

$x^2+y^2=r^2$x2+y2=r2, when the centre of the circle is at the origin
$\left(x-h\right)^2+\left(y-k\right)^2=r^2$(x−h)2+(y−k)2=r2, when the centre of the circle is not at the origin

We may be asked to calculate the:

Coordinates of the centre
The length of the radius
The equation of the circle

Graphs with Two Equations

To find the solution to two different equations, you can either solve them graphically or algebraically using simultaneous equations.

Worked Examples

Question 1

Consider the lines $L_1$L1, $y=-4x+5$y=−4x+5, and $L_2$L2, $y=x-1$y=x−1.

Find the midpoint $M$M of their $y$y-intercepts.

Express your answer in coordinate form ($x$x,$y$y)
Find the equation of the line that goes through the point $M$M and has slope $\frac{1}{3}$13. Express the equation in general form.

Question 2

Consider the function $y=x^2$y=x2

Complete the following table of values.

$x$x $-3$−3 $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $3$3

$y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$
Plot the points in the table of values.

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Hence plot the curve.

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Are the $y$y values ever negative?
No
A
Yes
B
Write down the equation of the axis of symmetry.
What is the minimum $y$y value?
For every $y$y value greater than $0$0, how many corresponding $x$x values are there?
$3$3
A
$1$1
B
$2$2
C

$x$`x`	$-3$−3	$-2$−2	$-1$−1	$0$0	$1$1	$2$2	$3$3
$y$`y`	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$	$\editable{}$

Question 3

A parabola has its turning point at $x=-3$x=−3 and one of the $x$x-intercepts is $x=1$x=1.

What is the $x$x value of the other $x$x-intercept?
If it has a $y$y-intercept at $3$3, write down the equation of the parabola. You may use the general factored form of $y=a\left(x-m\right)\left(x-n\right)$y=a(x−m)(x−n).
What are the coordinates of the turning point?

Turning point $=$=$\left(\editable{},\editable{}\right)$(,)

Question 4

The equation of a circle is given by $\left(x-4\right)^2+\left(y+1\right)^2=36$(x−4)2+(y+1)2=36.

Find the coordinates of the centre of this circle.
Find the radius of the circle.
Plot the graph for the given circle.

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Question 5

Consider the function $y=3^{-x}-1$y=3−x−1.

Find the $y$y-intercept of the curve $y=3^{-x}-1$y=3−x−1.
Find the horizontal asymptote of the curve $y=3^{-x}-1$y=3−x−1.
Now use your previous answers to plot $y=3^{-x}-1$y=3−x−1.

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Question 6

The following container is filled with water at a constant rate. Choose the graph that shows how the water level in the container is changing with time.