Investigating $y=ax^2+bx+c$

On the grid to the left you can see the graph of $y=2x^2-8x-2$. Looking at the graph you can see that the axis of symmetry of this parabola is the line $x=2$. Where is the vertex of this parabola? (Notice that the scale of the graph has again been expanded.)

Using the sliders, change the values of $a$, $b$, and $c$, and notice how each affects the shape and position of the graph of $y=ax^2+bx+c$.

When $c$ changes, does the graph get narrower and wider, or does it stay the same width?

When $c$ changes, does the graph move up and down, does it move left and right, or does it do both of those things?

When $b$ changes, does the graph get narrower and wider, or does it stay the same width?

When $b$ changes, does the graph move up and down, does it move left and right, or does it do both of those things?

When $b$ changes, there is one point that always stays on the graph. What is the $x$-coordinate of that point?

When $a$ changes, does the graph get narrower and wider, or does it stay the same width?

When $a$ changes, there is one point that always stays on the graph. What is the $x$-coordinate of that point?

Slide the slider for each of the coefficients $a$, $b$, and $c$ and observe what happens. Using your observations, note in the table below which coefficients affect the locations of the vertex, axis of symmetry, and roots of $y=ax^2+bx+c$.

	Does it affect the...
	vertex?	axis of symmetry?	roots?

The axis of symmetry of $y=ax^2+bx+c$

Use the sliders to change the values of $a$, $b$, and $c$, and answer the following questions.

What is the axis of symmetry of $y=x^2-4x$?

What is the axis of symmetry of $y=x^2-4x+2$?

What is the axis of symmetry of $y=x^2-4x-3$?

As you see, the value of $c$ doesn’t affect the axis of symmetry. This means that if you know that the axis of symmetry of $y=x^2+4x$ is the line $x=-2$, you also know that the axis of symmetry of $y=x^2+4x+3$ is the line $x=-2$.

As you learned in the previous lesson, the axis of symmetry of $y=ax^2+bx$ is the line $$x=-b/{2a}$$. What is the axis of symmetry of $y=ax^2+bx+c$?

What is the axis of symmetry of $y=x^2-2x+3.5$?

The vertex of $y=ax^2+bx+c$

Set $a=1$, $b=-4$, and $c=2$ to look at the graph of $y=x^2-4x+2$. Using the formula $$x=-b/{2a}$$, you can calculate that the axis of symmetry of this parabola is the line $x=2$. Also, notice that the vertex of this parabola is the point $(2,-2)$. Now slide $c$ to 4.5. The axis of symmetry of the parabola is still the line $x=2$, but the vertex has moved. The location of the vertex isn’t obvious from the graph, but you can find it algebraically:
1) The vertex is on the axis of symmetry, so its $x$-coordinate is 2.
2) The vertex is a point on the parabola, so it satisfies the equation for the parabola. This means that if we plug the $x$-coordinate of the vertex into the equation, we will get the $y$-coordinate:
$y=x^2-4x+4.5$
$x=2$ → $y=(2)^2-4(2)+4.5=4-8+4.5=0.5$
So the vertex of this parabola is the point $(2,0.5)$.

Use the method shown above to find the vertex of $y=x^2-2x+3.5$. Check your answer using the sliders and graph to the left.

Comparing $y=ax^2+bx+c$ to $y=a(x-h)^2+k$

So far you know that both $y=ax^2+bx+c$ and $y=a(x-h)^2+k$ have graphs which are parabolas. Let’s see which form gives you more information about the parabola and, knowing where the vertex of a parabola is, which form allows you to write an equation for the parabola more easily.

Here is the graph of $y=a(x-h)^2+k$. Use the sliders to change the values of $h$ and $k$ to remind yourself of what the values of $h$ and $k$ represent. Where is the vertex of the parabola $y=(x-2)^2+3$?

Find the vertex of the parabola $y=x^2+2x+3$ using the method from findVertex. (Scroll back to see findVertex.)

Click to get back a graph of $y=ax^2+bx+c$ with sliders for $a$, $b$, and $c$. Use the sliders and graph to check your answer.

Use the sliders to change the values of $a$, $h$, and $k$ and find an equation in the form $y=a(x-h)^2+k$ for a parabola with vertex at $(4,2)$.

Use the sliders to change the values of $a$, $b$, and $c$ to find an equation for a parabola in the form $y=ax^2+bx+c$ with vertex at $(4,2)$.

Converting $y=ax^2+bx+c$ to $y=a(x-h)^2+k$

A quadratic equation in the form $\cl"red"{y=ax^2+bx+c}$ is said to be in standard form, while an equation in the form $\cl"blue"{y=a(x-h)^2+k}$ is said to be in vertex form. In this section you will learn to convert equations such as $y=2x^2-2x+3$ from standard form to vertex form.

In both forms $a$ is the coefficient of $x^2$, so the $a$’s in both forms are the same. $h$ and $k$ are the $x$- and $y$-coordinates of the vertex, so if you find the vertex of $y=2x^2-2x+3$, you will know $h$ and $k$.

As you did in findVertex above, find the vertex of $y=2x^2-2x+3$.

What are the values of $a$, $h$ and $k$?

Click . You will see the graph of $y_1=2x^2-2x+3$ in red and sliders for $a$, $h$, and $k$. Change the values of $a$, $h$, and $k$ to the ones you found.

Once you’ve changed the values, do the two parabolas match? (When the two match, you will see only one parabola in blue.)

Write $y=2x^2-2x+3$ in $y=a(x-h)^2+k$ form.