The Vertex (Standard) Form Of A Quadratic Worksheet printable pdf download

Section 2.6B: The Vertex (Standard) Form of a Quadratic:

The other quadratic form that you will need to know is called the vertex form. For an arbitrary parabola, the highest (if

a < 0) or lowest (if a > 0) point is called the vertex. The vertex is generally a point labeled as (h, k)

The vertex form of a quadratic function is:

f (x) = a(x

h) + k

Example: Answer the following regarding quadratic equations.

1. Determine the vertex of the function f (x) = 5(x

3) + 9.

2. Determine an equation for the quadratic with vertex (4, 5) and goes through the point ( 2, 3).

3. Determine an equation for the quadratic with x-intercepts at

2 and 8, and has a low point of

Converting from Quadratic Form to Vertex Form:

For this unit, you will have to convert from Quadratic Form to Vertex form. Note that for Quadratic Form, our pertinent

coeﬃcients are a, b, and c. For the Vertex form, they are a, h, and k.

The a value for both forms are the same. To compute h and k, we have the following formulas:

h =

b/(2a)

k = f (h)

Examples: Put the following functions in standard (vertex) form.

f (x) = 2x

x + 1

g(x) =

2x + 9

h(x) = 5 + 3x

j(x) = x + 3x

More on the Vertex:

As deﬁned earlier, the vertex is either the highest or lowest point of a quadratic. As such, this is also a useful tool for

determining the range of a quadratic function. Here’s a recap:

Leading Coeﬃcient

Vertex Notes

Parabola Shape

Function Range

a is Positive

(h, k) is lowest point.

right-side up

[k,

)

a is Negative

(h, k) is highest point.

upside down

(

, k]

Word Problems Involving Quadratic Functions:

1. An object is thrown upward. The height, h, in feet, at time t, in seconds, is given by the formula h(t) =

16t + 96t.

(a) Determine the number of seconds required for it to hit the ground.

(b) Determine the maximum height of the object.

2. A diver jumps vertically oﬀ a diving board at time t = 0. The divers height h above the water (in feet), t seconds

later is given by the formula h(t) =

16t + 6t + 5.

(a) Determine the number of seconds t required for the diver to reach the water (h = 0)

(b) How many seconds after jumping is the diver at her maximum height above the water?

3. Let x represent the number of widgets sold, and p(x) the price per widget in dollars. The ﬁrm begins by selling x =

300 widgets at a set price of $45 each. After holding a ”sale”, the ﬁrm proposes that a $10 discount on the price will

yield an increase of 40 more widgets sold.

(a) Find the linear pricing function p(x) for this model.

(b) Based on the fact that the ﬁrm wants both sales and the price to be positive, what is the appropriate domain of

the pricing function p(x)?

(d) What is the price p that yields maximum revenue?

4. Farmer Brown has 400 yards of fencing with which to build a rectangular corral. He wants to divide it evenly into

three pens, so he adds in two divider fences, as shown below. Determine the maximum area enclosed by the corral.

5. Flights of leaping animals typically have parabolic paths. The ﬁgure below illustrates a frog jump superimposed on

a coordinate plane. The length of the leap is 9 feet, and the maximum height oﬀ the ground is 10.5 feet. Find a

standard equation (in the form y = a(x

h) + k) for the path of the frog.

6. A doorway has the shape of a parabolic arch and is 9 feet high at the center and 6 feet wide at the base. If a

rectangular box 5 feet high must ﬁt through the doorway, what is the maximum width the box can have?

The Vertex (Standard) Form Of A Quadratic Worksheet

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