How much information do we need to be able to graph a quadratic function? One way is to find the coordinates of the $x$x-intercepts, which we can then use to find the coordinates of the vertex. Our first step to the $x$x-intercepts will be to factor the equation.
There are many techniques for factoring quadratics, these are covered in the factoring section. We can factor by:
- Using the greatest common factor
- Factor special forms such as the difference of two squares or perfect squares
- Grouping in pairs
Finding zeros
Recall that if we have two factors, like $a$a and $b$b, and we multiply them together so that they equal $0$0, then one of those factors ($a$a or $b$b) must be $0$0. A written solution to a question like this would be similar to the following:
If $a\times b=0$a×b=0 then $a=0$a=0 or $b=0$b=0.
Now, let's see why this is so useful when graphing quadratics.
$x$x-intercepts occur when the value of $y$y is zero
So finding the $x$x-intercepts is the same as solving the quadratic. Refer to the lesson on Physical characteristics of solving an equation for more detail. As we saw when Solving by Factoring, once fully factored, we set each factor to zero and solve to get the values of the zeros. These values correspond to the x-coordinates of the x-intercepts of the graph of the quadratic function.
Worked example
Find the $x$x-intercepts of the function $y=2x\left(x+6\right)$y=2x(x+6).
Think: Finding the $x$x-intercepts of this function is the same as finding the roots of the equation $2x\left(x+6\right)=0$2x(x+6)=0.
Do: Set each factor to zero, so either
| $2x$2x | $=$= | $0$0 | or | $x+6$x+6 | $=$= | $0$0 |
| $x$x | $=$= | $0$0 | $x$x | $=$= | $-6$−6 |
This curve crosses the $x$x-axis at $0$0 and $-6$−6.
The vertex
All parabolas (the shape of a quadratic function) are symmetric functions. The axis of symmetry of a parabola passes through its turning point. Since parabolas have only one turning point, this point is given a special name: the vertex. This is the maximum or minimum value of a quadratic function.
The vertex always occurs midway between the two roots, and lies on the axis of symmetry.
In the example above where we had $x$x-intercepts of $0$0 and $-6$−6, the $x$x-coordinate of the vertex will occur halfway between $0$0 and $-6$−6.
In some cases you can tell by inspection what the midway point is. In this case we can see that it is $-3$−3. If we can't tell by inspection we can just find the average:
| $\frac{x_1+x_2}{2}$x1+x22 | $=$= | $\frac{0+\left(-6\right)}{2}$0+(−6)2 |
| $=$= | $\frac{-6}{2}$−62 | |
| $=$= | $-3$−3 |
| $f(x)$f(x) | $=$= | $2x\left(x+6\right)$2x(x+6) |
| $f(3)$f(3) | $=$= | $2\times\left(-3\right)\times\left(-3+6\right)$2×(−3)×(−3+6) |
| $=$= | $-6\times3$−6×3 | |
| $=$= | $-18$−18 |
So the vertex has the coordinates $\left(-3,-18\right)$(−3,−18).
Graphing the quadratic
Now that we know that the function $y=2x\left(x+6\right)$y=2x(x+6) has $x$x-intercepts at $x=0$x=0 and $x=-6$x=−6, and has a vertex at $\left(-3,-18\right)$(−3,−18), we can graph the parabola that passes through these three points. The parabola looks like this:
Practice questions
Question 1
Consider the parabola $y=x\left(x+6\right)$y=x(x+6).
Find the $y$y value of the $y$y-intercept.
Find the $x$x values of the $x$x-intercepts.
Write all solutions on the same line separated by a comma.
State the equation of the axis of symmetry.
Find the coordinates of the turning point.
Turning point $=$=$\left(\editable{},\editable{}\right)$(,)
Plot the graph of the parabola.
Loading Graph...
Question 2
Consider the parabola $y=\left(x+1\right)\left(x-3\right)$y=(x+1)(x−3).
Find the $y$y value of the $y$y-intercept.
Find the $x$x values of the $x$x-intercepts.
Write all solutions on the same line separated by a comma.
State the equation of the axis of symmetry.
Find the coordinates of the vertex.
Vertex $=$=$\left(\editable{},\editable{}\right)$(,)
Graph the parabola.
Loading Graph...
Question 3
Consider the equation $y=x^2-6x+8$y=x2−6x+8.
Factor the expression $x^2-6x+8$x2−6x+8.
Hence or otherwise find the $x$x values of the $x$x-intercepts of the quadratic function $y=x^2-6x+8$y=x2−6x+8. Write all solutions on the same line separated by a comma.
Find the $x$x value of the turning point.
Find the $y$y value of the turning point.
Plot the graph of the function.
Loading Graph...
Outcomes
10D.QR3.03
Determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts (i.e., the zeros) of the graph of the corresponding quadratic relation, expressed in the form y = a(x – r)(x – s);