Task Sets: Use factoring and completing the square - Common Core Math - HS Functions

A

Recall / Warm-Up

1.

Which factored form is correct for $f(x) = x^2 - 7x + 12$ ?

A.

$(x - 3)(x - 4)$

B.

$(x + 3)(x + 4)$

C.

$(x - 6)(x - 2)$

D.

$(x - 3)(x + 4)$

2.

What is the vertex of $f(x) = (x - 5)^2 + 3$ ?

A.

$(-5, 3)$

B.

$(5, 3)$

C.

$(5, -3)$

D.

$(0, 3)$

3.

For the parabola $f(x) = -2(x - 1)^2 + 7$ , does the vertex represent a maximum or minimum value?

A.

Maximum, because $a = -2 < 0$ means the parabola opens downward.

B.

Minimum, because $a = -2 < 0$ means the parabola opens downward.

C.

Maximum, because the vertex $y$ -value is positive.

D.

Minimum, because the vertex $y$ -value is $7 > 0$ .

B

Fluency Practice

1.

Factor $f(x) = x^2 + x - 12$ and identify its zeros.

A.

Zeros at $x = 3$ and $x = -4$

B.

Zeros at $x = -3$ and $x = 4$

C.

Zeros at $x = 12$ and $x = -1$

D.

No real zeros

2.

Factor $g(x) = x^2 - 6x - 27$ . Enter the two zeros separated by a comma (smaller value first).

3.

Complete the square for $h(x) = x^2 + 6x + 2$ .

Step 1: Half of 6 is $\_\_\_$ ; square it to get $\_\_\_$ .
Step 2: Rewrite: $(x + \_\_\_)^2 - \_\_\_ + 2 = (x + 3)^2 + \_\_\_$ .

half of 6:

half squared:

value inside parentheses:

value subtracted:

final constant:

4.

Complete the square to rewrite $f(x) = x^2 - 10x + 18$ in vertex form. Enter the $y$ -coordinate of the vertex.

5.

The function $f(x) = -3(x - 2)^2 + 12$ is in vertex form. What is the maximum value of $f$ ?

A.

$-3$

B.

$2$

C.

$12$

D.

There is no maximum.

C

Varied Practice

D

Word Problems

1.

A ball is launched upward with height modeled by $h(t) = -16t^2 + 64t + 5$ , where $h$ is in feet and $t$ is in seconds.

1.

Complete the square to find the vertex. What is the maximum height the ball reaches, in feet?

2.

At what time $t$ (in seconds) does the ball reach its maximum height?

A.

$t = 69$ seconds

B.

$t = 5$ seconds

C.

$t = 2$ seconds

D.

$t = 64$ seconds

2.

A company's weekly profit (in hundreds of dollars) is modeled by $P(x) = -2x^2 + 20x - 42$ , where $x$ is the number of units sold. Factor $P(x)$ to find the break-even points (where profit = 0).

Enter the two break-even $x$ -values separated by a comma (smaller value first).

E

Error Analysis

1.

A student completed the square for $f(x) = x^2 - 8x + 3$ as follows:

Half of $-8$ : $-4$ . Square: $16$ .
$f(x) = (x^2 - 8x + 16) + 3 = (x - 4)^2 + 3$ .

The student concluded: vertex is $(4, 3)$ .

What error did the student make?

A.

The student computed half of $-8$ incorrectly; it should be $-16$ , not $-4$ .

B.

The student added $16$ inside the expression but forgot to subtract $16$ to keep the function equal. The correct vertex form is $(x-4)^2 - 13$ .

C.

The student used the wrong sign — the vertex should be at $(-4, 3)$ .

D.

The student's work is correct — the vertex is $(4, 3)$ .

2.

A student looked at $g(x) = (x + 4)^2 - 9$ and wrote:

"The vertex is at $(-4, -9)$ and the axis of symmetry is $x = -4$ ."

Another student disagreed: "The vertex is at $(4, -9)$ and the axis of symmetry is $x = 4$ ."

Which student is correct, and why?

A.

The first student is correct. The sign inside gives $h = -4$ , so the axis is $x = -4$ .

B.

The second student is correct. The $+4$ gives $h = 4$ , so the vertex is $(4, -9)$ .

C.

Both are wrong — the axis is $x = 0$ for all parabolas.

D.

The vertex is $(4, 9)$ and the axis is $x = 4$ .

F

Challenge / Extension

1.

A farmer has 120 feet of fencing to enclose a rectangular garden against a barn wall (the barn serves as one side, so only three sides need fencing). If the width of the garden (perpendicular to the barn) is $x$ feet, the area is $A(x) = x(120 - 2x)$ .

Complete the square (or use the vertex formula) to find the maximum area in square feet.

2.