Exercises: Choose equivalent forms to reveal properties - Common Core Math - HS Algebra

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

Which expression is equivalent to $(2^3)^4$ ?

A.

$2^7$

B.

$2^{12}$

C.

$8^4$

D.

$2^{34}$

2.

Which pair of factors correctly factors $x^2 - 5x + 6$ ?

A.

$(x - 1)(x - 6)$

B.

$(x + 2)(x + 3)$

C.

$(x - 2)(x - 3)$

D.

$(x - 6)(x + 1)$

3.

For the quadratic $f(x) = x^2 - 6x + 8$ , what is the value of $f(0)$ ? (This is the $y$ -intercept.)

B

Fluency Practice

Apply the procedures directly. Show all steps.

$Table showing the three forms of a quadratic and what each form reveals: standard reveals y-intercept, factored reveals zeros, vertex reveals the vertex and max/min.$

1.

Factor $f(x) = x^2 - 7x + 10$ and find its zeros. Enter the larger zero.

2.

Factor $g(x) = 2x^2 + x - 6$ and find its zeros. Enter the positive zero as a fraction (e.g., 3/2).

3.

Factor $h(x) = -x^2 + 6x - 8$ and find its zeros. Enter the sum of the two zeros.

4.

Complete the square to write $f(x) = x^2 + 6x + 5$ in vertex form. What is the vertex?

A.

$(-3, -4)$

B.

$(3, -4)$

C.

$(-3, 14)$

D.

$(-6, 5)$

5.

Complete the square to write $f(x) = 2x^2 - 12x + 7$ in vertex form $a(x-h)^2 + k$ .
What is the minimum value $k$ ?

6.

An account earns 8% per year, modeled by $A(t) = (1.08)^t$ where $t$ is in years.
Which expression shows the equivalent quarterly growth factor?

A.

$(1.08)^{t/4}$

B.

$(1.02)^{4t}$

C.

$(1.08^{1/4})^{4t}$

D.

$(1.08^4)^{t/4}$

C

Varied Practice

Problems appear in different formats. Read each carefully.

D

Word Problems

Set up each problem carefully. Identify which form of the expression you need.

1.

A company's weekly profit (in thousands of dollars) is modeled by
$P(x) = -2x^2 + 10x - 8$ , where $x$ is the number of units produced (in hundreds).

At what production levels (in hundreds of units) does the company break even?
Enter the larger break-even value.

2.

A ball is kicked from ground level and its height (in feet) after $t$ seconds is modeled by
$h(t) = -16t^2 + 64t$ .

1.

Factor $h(t)$ to find when the ball hits the ground ( $h = 0$ ). Enter the positive time value (in seconds).

2.

Complete the square (or use the vertex of the parabola) to find the maximum height of the ball in feet.

3.

The height of a projectile (in feet) is modeled by $h(t) = -16t^2 + 96t + 112$ ,
where $t$ is time in seconds.

1.

Complete the square to find the maximum height of the projectile in feet.

2.

At what time (in seconds) does the projectile reach its maximum height?

4.

A savings account compounds interest monthly. The balance after $t$ years is
$B(t) = 1000 \cdot (1.12)^t$ , where 12% is the annual growth rate.

1.

Which expression correctly rewrites $B(t)$ to reveal the monthly growth factor?

A.

$1000 \cdot (1.01)^{12t}$

B.

$1000 \cdot (1.12^{1/12})^{12t}$

C.

$1000 \cdot (1.12)^{12t}$

D.

$1000 \cdot (1.12^{12})^{t/12}$

2.

To the nearest hundredth, what is the approximate monthly growth factor?
Use a calculator to evaluate $1.12^{1/12}$ .

E

Error Analysis

Each problem shows student work with a mistake. Identify and explain the error.

1.

Maya completed the square for $f(x) = x^2 - 10x + 3$ :

Group: $(x^2 - 10x) + 3$
Half of $-10$ is $-5$ ; $(-5)^2 = 25$
Add 25: $(x^2 - 10x + 25) + 3$
Factor: $(x - 5)^2 + 3$
Conclusion: vertex is $(-5, 3)$

Maya made two errors. Which choice correctly identifies both?

A.

Maya forgot to subtract 25 to compensate, making $k = 3$ wrong; and she misread the vertex sign, giving $x = -5$ instead of $x = 5$ .

B.

Maya squared incorrectly; $(-5)^2$ should equal $-25$ , not $25$ .

C.

Maya chose the wrong form; she should have factored instead of completing the square.

D.

Maya correctly completed the square; the vertex is $(-5, 3)$ .

2.

Jordan completed the square for $g(x) = 2x^2 - 8x + 5$ :

Factor out 2: $g(x) = 2(x^2 - 4x) + 5$
Half of $-4$ is $-2$ ; $(-2)^2 = 4$
Add 4 inside: $g(x) = 2(x^2 - 4x + 4) + 5 - 4$
Factor: $g(x) = 2(x - 2)^2 + 1$
Minimum value: $1$

What error did Jordan make in step 3, and what is the correct minimum value?

A.

Jordan subtracted 4 instead of adding 4 outside; the minimum is $-3$ .

B.

Jordan subtracted 4 outside but should have subtracted $2 \times 4 = 8$ ; the minimum is $-3$ .

C.

Jordan should have added 4 outside instead of subtracting 4; the minimum is $9$ .

D.

Jordan made no error; the minimum value is 1.

F

Challenge / Extension

These problems are for extension. They require multi-step reasoning.

1.

The function $f(x) = x^2 - 6x + 8$ can be written as $(x-2)(x-4)$ or as $(x-3)^2 - 1$ .

Explain: (a) what each form reveals about the graph of $f$ , and (b) why neither the zeros nor
the vertex is visible from the standard form $f(x) = x^2 - 6x + 8$ .

2.

A colony of bacteria doubles every 5 hours. Its population is modeled by
$Q(t) = Q_0 \cdot 2^{t/5}$ , where $t$ is time in hours.