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

In the last lab, 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 $y=2x^2-8x-2$.

Question 1. In the grid on the left you can see the graph of $y=2x^2-8x-2$. (The equation for the graph is written below the grid.) 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?

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

What happens to the graph when $c$ changes? (If the graph moves, say in which direction; if its shape changes, say how.)

What happens to the graph when $b$ changes? Is there a point that always remains on the graph? What is this point?

What happens to the graph when $a$ changes? (How does $a$ affect the shape? What point on the graph doesn't move when you change $a$?)

Question 3. Which of $a$, $b$, and $c$ affect the location of the vertex?

Which of $a$, $b$, and $c$ affect the axis of symmetry?

Which of $a$, $b$, and $c$ affect the location of the roots?

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

Question 4. 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 $y=x^2-4x-3$?

Question 5. 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 $x=-2$.

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 $(2,0.5)$.

Question 6. 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.

Question 7. Click on to see 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 $y=(x-2)^2+3$?

Question 8. Find the vertex of the parabola $y=x^2+2x+3$ using the method from Question 6. Click on so you can use the sliders and graph to the left to check your answer.

Question 9. Click on . 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 $(4,2)$.

$y=$$\,(\,$$\,)^2$

Question 10. Click on . 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 $(4,2)$.

$y=$ $x^2$

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

In this section you will learn to rewrite equations such as $y=x^2-x+3$ in vertex form (the form $y=a(x-h)^2+k$).

Question 11. 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 $y=x^2-x+3$, you will know $h$ and $k$.

As you did in Question 6, find the vertex of $y=x^2-x+3$.

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

$a=$

$h=$

$k=$

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

$y=$$\,(\,$$\,)^2$

To check your answer, click on . You will see the graph of $y_1=x^2-x+3$ in red and sliders for $a$, $h$, and $k$. Change the values of $a$, $h$, and $k$ to the ones you found and see if the two parabolas match. (When the two match, you will see only one parabola in blue.)

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