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

In the last lesson, you learned about the shapes of the graphs and other properties of equations such as \$y=2x^2-8x\$. Now, you will learn about slightly different equations such as \$y=2x^2-8x-2\$.

 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.