### Student Page4.6.5Forms of Quadratic Functions Card Sort A4US

There are three forms of a quadratic function. Each one is important in its own way. Each one makes one or more aspects of a quadratic function evident. The forms are:

\begin{equation*} \text{Standard: } ax^2 + bx + c \qquad \text{Vertex: } a(x - h)^2 + k \qquad \text{Factored: } y = a(x - p)(x - q) \end{equation*}

#### 1.

A card sort and recording sheet are included below. Cut the cards apart.

##### (a)

Sort the cards to find pairs of equations that represent the same quadratic function. Write the equations in the table below.

##### (b)

Verify algebraically that the cards you have matched name the same function. Show your work! (Check the graphs, too!)

#### 2.

##### (a)

Find the vertex and $$x$$-intercepts for each card set. List them as ordered pairs in the table.

##### (b)

One equation form is missing from each set. Determine the missing form and write the missing equation (replace letters with appropriate numbers) in the table below. Verify algebraically that the missing equation matches the other equations in the set.

#### 3.

What features of the graph of a quadratic function are evident from:

##### (a)

The standard form of the equation?

##### (b)

The vertex form of the equation?

##### (c)

The factored form of the equation?

#### 4.

##### (a)

Find all three forms of a quadratic function with $$x$$-intercepts at $$x = 2$$ and $$x = -1.5\text{.}$$

##### (b)

Is there only one function that satisfies the conditions in Task 4.6.5.4.a? How do you know?

##### (c)

Find an equation for a quadratic function that satisfies the conditions in Task 4.6.5.4.a and contains the point $$(3, 6)\text{.}$$ How many such quadratic functions are there?

#### 5.

##### (a)

A quadratic function has vertex $$(2, 4)$$ and an $$x$$-intercept at $$x = 0\text{.}$$ Where is the other $$x$$-intercept? How do you know?

##### (b)

Find an equation for a quadratic function that fits the conditions in Task 4.6.5.5.a. Show that your equation works.