Graphing Quadratic Equations Using Transformations

In the previous section we learned that the shape of [latex]y=x^2[/latex] is called a parabola. The turning point on the graph is called the vertex. The vertical line that passes through the vertex and splits the parabola into two mirror images is called the line of symmetry. All quadratic functions have graphs in the shape of a parabola.  However, in the general equation [latex]y=ax^2+bx+c[/latex] the values of [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] change the direction, shape, and position of the graph.  Let’s discover how changing these values can transform a graph.

How [latex]a[/latex] affects the graph of [latex]y=ax^{2}+bx+c[/latex]

Figure 1 shows an animation of the graph [latex]y=ax^2[/latex] as the value of [latex]a[/latex] moves between [latex]-10[/latex] and [latex]10[/latex].  Click on the desmos logo at the bottom right corner of the graph to manipulate the value of [latex]a[/latex].

Figure 1: Animation of [latex]y=ax^2[/latex] as [latex]a[/latex] changes.

The value of [latex]a[/latex] tells us whether the parabola opens upwards ([latex]a>0[/latex]) or downwards ([latex]a<0[/latex]). If [latex]a[/latex] is positive, the vertex is the turning point and the lowest point on the graph and the graph opens upward.  If [latex]a[/latex] is negative, the vertex is the turning point and the highest point on the graph and the graph opens downward.

Figure 2. Graph of [latex]y=ax^2[/latex] with different values of [latex]a[/latex].

The value of [latex]a[/latex] also tells us about the width of the graph.  When [latex]|a|>1[/latex], as in [latex]y=2x^2[/latex] in figure 2, the graph will appear more narrow than [latex]y=x^2[/latex].  When [latex]|a|<1[/latex], as in [latex]y=-\frac{3}{4} x^2[/latex] in figure 2, the graph will appear wider than  [latex]y=x^2[/latex].

Example

Match each function with its graph.

1. [latex]\displaystyle y=3{{x}^{2}}[/latex]

2. [latex]\displaystyle y=-2{{x}^{2}}[/latex]

3. [latex]\displaystyle y=-8{x}^{2}[/latex]

4. [latex]\displaystyle y=7{x}^{2}[/latex]

Graph A

Graph B

Graph C

Graph D

Solution

1. [latex]\displaystyle y=3{{x}^{2}}[/latex] opens upwards and passes through the point [latex](1, 3)[/latex]: Graph D

 

2. [latex]\displaystyle y=-2{{x}^{2}}[/latex]nopens downwards and passes through the point [latex](1, -2)[/latex]: Graph B

 

3. [latex]\displaystyle y=-8{x}^{2}[/latex] opens downwards and passes through the point [latex](1, -8)[/latex]: Graph C

 

4. [latex]\displaystyle y=7{x}^{2}[/latex] opens upwards and passes through the point [latex](1, 7)[/latex]: Graph A

 

We can compare the graphs to the graph of the parent equation [latex]y=x^2[/latex]. If [latex]|a|>1[/latex], the graph of [latex]y=ax^2[/latex] will be thinner than that of [latex]y=x^2[/latex]. If [latex]0<|a|<1[/latex], the graph of [latex]y=ax^2[/latex] will be broader than that of [latex]y=x^2[/latex]. And remember that when [latex]a<0[/latex] the graph opens downwards.

Example

Given the graph of [latex]y=x^2[/latex] shown in black, determine the relative value of [latex]a[/latex] in the graph of [latex]y=ax^2[/latex] shown in blue.

Graph 1

Graph 2

Graph 3

Graph 1. The blue graph is narrower than the graph of [latex]y=x^2[/latex] and opens upwards, which means that [latex]a>1[/latex].

 

Graph 2. The blue graph opens downwards so has a negative value of [latex]a[/latex].  It is also broader that [latex]y=x^2[/latex] so [latex]|a|<0[/latex]. This means that [latex]-11[/latex].

How [latex]c[/latex] affects graphs of quadratic equations

When a quadratic equation is in the form [latex]y=ax^2+c[/latex], [latex]b=0[/latex] and there is no [latex]x[/latex]-term. Figure 3 shows an animation of the graph [latex]y=x^2+c[/latex] as the value of [latex]c[/latex] moves between [latex]-10[/latex] and [latex]10[/latex].  Click on the desmos logo at the bottom right corner of the graph to manipulate the value of [latex]c[/latex].

Figure 3: Animation of [latex]y=x^2+c[/latex] as [latex]c[/latex] changes.

Changing the value of [latex]c[/latex] moves the parabola up or down the [latex]y[/latex]-axis. If [latex]c>0[/latex], the graph of [latex]y=x^2[/latex] moves up the axis [latex]c[/latex] units. If [latex]c<0[/latex], the graph of [latex]y=x^2[/latex] moves down the axis [latex]c[/latex] units.

Figure 4. Graph of [latex]y=x^2+c[/latex] with different values of [latex]c[/latex].

Try It

Determine the equation of each graph, using the graph of [latex]y=x^2[/latex] shown in black as the parent graph.

Graph 1

Graph 2

 

Show Answer

Graph 1: [latex]Graph 2: [latex]y=x^2+5[/latex]

When we are graphing any quadratic equation, it is also useful to know that [latex]c[/latex] corresponds to the [latex]y[/latex]-intercept of the graph of any quadratic equation.  The [latex]y[/latex]-intercept occurs when [latex]x=0[/latex] and is the point at which the graph crosses the [latex]y[/latex]-axis.  Recall that we determine the value of [latex]y[/latex] by substituting [latex]x=0[/latex] into the equation:

[latex]y=a(0)^2+b(0)+c=c[/latex]

For a quadratic equation, the [latex]y[/latex]-intercept is always the point [latex](0,c)[/latex].

How [latex]b[/latex] affects graphs of quadratic equations

When a quadratic equation is in the form [latex]y=ax^2+bx[/latex], [latex]c=0[/latex] and therefore the [latex]y[/latex]-intercept will always be [latex](0, 0)[/latex]. Figure 5 shows an animation of the graph [latex]y=x^2+bx[/latex] as the value of [latex]b[/latex] moves between [latex]-10[/latex] and [latex]10[/latex].  Click on the desmos logo at the bottom right corner of the graph to manipulate the value of [latex]b[/latex].

Figure 5. Animation of [latex]y=x^2+bx[/latex] as [latex]b[/latex] changes.

Changing [latex]b[/latex] moves the vertex and the line of symmetry, which is the vertical line that passes through the vertex of the parabola. When [latex]b>0[/latex] the vertex moves to the left. When [latex]b<0[/latex] the vertex moves to the right.  However, the vertex doesn’t just move horizontally, but vertically as well. Notice that the vertex moves following a parabolic curve!  Moving the vertex also moves the line of symmetry, since the line of symmetry passes through the vertex.  In addition, notice that the [latex]x[/latex]– and [latex]y[/latex]-intercepts stay the same at [latex](0, 0)[/latex], but the other [latex]x[/latex]-intercept moves between [latex]x=-10[/latex] when [latex]b=10[/latex] to [latex]x=10[/latex] when [latex]b=-10[/latex].

To help explain this movement, let’s find the [latex]x[/latex]-intercepts by setting [latex]y=0[/latex]:

[latex]\begin{equation}\begin{aligned} y&=x^2+bx \\ 0&=x^2+bx \;\;\quad\quad\quad\quad\text{Substitute }x=0 \\ 0&=x(x+b)\;\qquad\quad\quad\text{Factor out the GCF }x\\ x&=0\text{ or }x+b=0\quad\quad\text{Set each equation to zero}\\ x&=0\text{ or }x=-b\quad\quad\quad\text{Solve each equation}\end{aligned}\end{equation}[/latex]

So the [latex]x[/latex]-intercepts are [latex](0, 0)\text{ and }(0, -b)[/latex].  [latex](0, 0)[/latex] is a fixed point so does not move as the value of [latex]b[/latex] changes. However, [latex](0, -b)[/latex] changes as [latex]b[/latex] changes.  For example, when [latex]b=2[/latex] the intercept changes to [latex](0, -2)[/latex] and when [latex]b=-5[/latex] the intercept changes to [latex](0, 5)[/latex]. Do you see the pattern?

Figure 6. Graph of [latex]y=x^2+bx[/latex] with [latex]x[/latex]-intercepts at [latex](0, 0)[/latex] and [latex](-b, 0)[/latex].

Because of symmetry, the vertex is always exactly halfway between the two [latex]x[/latex]-intercepts [latex](0, 0)[/latex] and [latex](0, –b)[/latex].  For example, in the graph [latex]y=x^2+4x[/latex] the [latex]x[/latex]-intercepts are [latex](0, 0)[/latex] and [latex](0, -4)[/latex] (figure 7). The axis of symmetry is the vertical line that passes through the vertex and lies exactly halfway between [latex]x=-4[/latex] and [latex]x=0[/latex]. i.e. the line of symmetry is [latex]x=-2[/latex].

Figure 7. [latex]y=x^2+4x[/latex]

This also means that the vertex has an [latex]x[/latex]-value of [latex]-2[/latex] since the line of symmetry passes through the vertex. The [latex]y[/latex]-coordinate of the vertex is found by substituting [latex]x=-2[/latex] into the equation [latex]y=x^2+4x[/latex]:  [latex]y=(-2)^2+4(-2)=4-8=-4[/latex]. So the vertex is the point [latex](-2, -4)[/latex] as can be seen on the graph in figure 7.

In the more general case of [latex]y=x^2+bx[/latex], the line of symmetry is exactly halfway between the two [latex]x[/latex]-intercepts [latex](0, 0)[/latex] and [latex](0, -b)[/latex]. i.e. the line of symmetry is [latex]x=\frac{-b}{2}[/latex]. Of course, this assumes that [latex]a=1[/latex] and [latex]c=0[/latex]. If [latex]a[/latex] and [latex]c[/latex] take on different values, determining the exact movement of the graph becomes a little more difficult.

The following animation shows what happens to the graph of [latex]y=ax^2+bx+c[/latex] when the values of [latex]a, b[/latex] and [latex]c[/latex] are changed. Click on the desmos logo at the bottom right corner of the graph to manipulate the values.

Figure 8. Animation of [latex]y=ax^2+bc+c[/latex] as [a, b[/latex] and [latex]c[/latex] change.