# Investigating \$y=a(x-h)^2+k\$

In this lesson you will learn about graphs of equations of the form \$y=a(x-h)^2+k\$. For example, you will look at equations such as \$y=3x^2\$, \$y=-2x^2\$, and \$y=2(x+1)^2+3\$, and compare them to \$y=x^2\$. You will also learn about roots of quadratic equations and how the values of \$a\$, \$h\$, and \$k\$ affect the number of roots.

## Comparing \$y=x^2\$ to \$y=3x^2\$

Complete this table of values for the equations \$y_1=x^2\$ and \$y=3x^2\$. Note that \$3x^2\$ means \$3(x^2)\$ not \$(3x)^2\$: in an expression without parentheses, you raise a base to a power before multiplying it by another number. For example, when \$x=2\$, \$3x^2=3(2^2)=3(4)=12\$.
\$x\$ \$y_1\$ \$y\$

Using the table above, find the vertex and axis of symmetry of each parabola.

Equationvertexaxis of
symmetry
\$y_1=x^2\$ \$x=0\$
\$y=3x^2\$
 Click to look at the graphs of \$y_1=x^2\$ and \$y=3x^2\$. Which is narrower: the red parabola (\$y_1\$) or the blue parabola (\$y\$)?

Here is the graph of \$y=ax^2\$. Use the slider to change the value of \$a\$ and answer the following questions.

 As \$a\$ increases, does the graph get narrower or wider?
 As \$a\$ increases, does the axis of symmetry of the graph move left, move right, or stay in the same place?
 As \$a\$ increases, does the vertex of the graph move left, move right, move up, move down, or stay in the same place?
 Put \$y=0.5x^2\$, \$y=3x^2\$, \$y=0.25x^2\$, \$y=7x^2\$, and \$y=2x^2\$, in order from widest to narrowest.

## What happens when \$a\$ is negative?

Here are the graphs of \$y_2=2x^2\$ and \$y=-2x^2\$.

 Is the graph of \$y=-2x^2\$ narrower than the graph of \$y_2=2x^2\$, wider than that graph, or the same width as that graph?
 Do \$y_2\$ and \$y\$ point in the same direction, or different directions?
 If you flip the graph of \$y_2\$ across one of the coordinate axes, you will get the graph of \$y\$. Which coordinate axis is it? The -axis
 What is the largest value of \$y\$?
 Can \$y\$ ever be positive?

Complete this table.

Equationvertexaxis of
symmetry
\$y_2=2x^2\$
\$y=-2x^2\$

Here is the graph of \$y=ax^2\$. Use the slider to change the value of \$a\$ to answer the following questions.

 We say that the graphs of \$y=0.25x^2\$, \$y=2x^2\$, and \$y=3x^2\$ open up. In what direction do the graphs of \$y=-2x^2\$, \$y=-0.5x^2\$, and \$y=-3x^2\$ open?
 Is the graph of \$y=-3x^2\$ narrower than the graph of \$y=2x^2\$, wider than that graph, or the same width as that graph?

## The vertex and axis of symmetry of \$y=a(x-h)^2+k\$

Here is the graph of \$y=a(x-h)^2+k\$. Use the sliders to change the values of \$a\$, \$h\$, and \$k\$ so that you are looking at the graph of \$y=2(x-1)^2+2\$. Fill in the following table. (The equation for the graph is written below the grid.)

Equation\$a\$\$h\$\$k\$vertexaxis of
symmetry
 Where is the vertex of the graph of \$y=a(x-h)^2+k\$?
 What is the axis of symmetry of the graph of \$y=a(x-h)^2+k\$?
 As \$a\$ increases from \$0\$ to \$5\$, does the graph of \$y\$ become narrower or wider?
 As \$a\$ decreases from \$0\$ to \$-5\$, does the graph of \$y\$ become narrower or wider?
 If \$a\$ is positive, does the graph of \$y\$ open up or down?
 If \$a\$ is negative, does the graph of \$y\$ open up or down?

## Roots of quadratic equations

Change the values of \$a\$, \$h\$ and \$k\$ so that you are looking at the graph of \$y=-3(x+1)^2+3\$.

Fill in the following table with how many times the graph of each equation meets the \$x\$-axis.

EquationNumber of times it
meets the \$x\$-axis

The points where the graph of an equation meets the \$x\$-axis are the \$x\$-intercepts of the equation. We call the \$x\$-coordinates of these points the roots of the equation.

You have seen that \$y=-3(x+1)^2+3\$ has 2 roots and \$y=2(x+1)^2+3\$ has no roots. Use the sliders to set \$a=2\$ and \$h=1\$. Use the slider for \$k\$ to change its value (without changing \$a\$ or \$h\$).

 For what values of \$k\$ does the equation have 2 roots?
 1 root?
 No roots?
 Does the equation ever have more than 2 roots?
 Does changing \$h\$ affect the number of roots of \$y=a(x-h)^2+k\$?

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\$ that has each possible number of roots.

Complete this table to summarize the effects of \$a\$ and \$k\$ on the number of roots of \$y=a(x-h)^2+k\$.

ConditionNumber
of roots