Alg2 S2 Midterm Review

Alg2 Midterm Review

(semester 2)​

Exponential functions require two things:

-> a y-intercept (initial value)

-> a growth factor (multiplier)

Their equations are usually formatted the same way -->​

Exponential functions might use y OR f(x) notation

Exponential Functions

Multiple Choice

In the equation $$y=6\left(1.4\right)^x$$   what is the y-intercept?

• 0

• 1.4

• 6

• 8.4

Multiple Choice

In the equation $$y=3.2\cdot8^x$$  what is the growth factor?

• 0

• 8

• 25.6

• 3.2

Multiple Choice

Which equation has a y-intercept of 4.5 and a growth factor of 6.7.

• $$y=4.5\left(6.7\right)^x$$  

• $$y=6.7\left(4.5\right)^x$$  

• $$y=4.5+6.7x$$  

• $$y=6.7+4.5x$$  

Multiple Choice

Benji decides to start a piggy bank. He deposits $0.30 at the start, and then doubles the amount he adds on each following day. Which equation represents A, the amount Benji adds to the piggy bank on day t?

• $$A=2\left(0.3\right)^x$$  

• $$A=2\left(0.3\right)^t$$  

• $$A=0.3\left(2\right)^x$$  

• $$A=0.3\left(2\right)^t$$  

Multiplying by 1 (100%) keeps a value the same.

If a value is changing by a percent increase or decrease, we can add or subtract the percentage from 100% to get our new multiplier (our decimal growth factor).​

Divide by 100 to get the decimal version of a percent.

Decimal Growth Factors

​

Multiple Choice

Which growth factor represents a 14% decrease?

• 86

  because

  100% – 14% = 86%

• 0.86

  because

  100% – 14% = 86%

  and 86% is 0.86

• 114

  because

  100% + 14% = 114%

• 1.14

  because

  100% + 14% = 114%

  and 114% is 1.14

Multiple Choice

Which exponential equation represents a 67% increase?

• $$y=4\left(1.67\right)^x$$  

• $$y=8\left(167\right)^x$$  

• $$y=3.5\left(0.67\right)^x$$  

• $$y=9\left(67\right)^x$$  

Slope (also called the unit rate or rate of change) can ​be found by finding the rise (difference in y-values) divided by the run (difference in x-values) between two points.

Slopes are usually written as fractions, not decimals.

Finding Slopes

Multiple Choice

Which expression could be used to find the slope

between the points (1, 3) and (-4, 2)?

• $$\frac{3-2}{4-1}$$  

• $$\frac{3-2}{1-4}$$  

• $$\frac{3-2}{1-\left(-4\right)}$$  

• $$\frac{3+2}{1+4}$$  

Multiple Choice

Marly used the expression $$\frac{-3-9}{5-\left(-6\right)}$$  to find the slope of a line m.

Based on Marly's work, which two points do we know are on line m?

• (-3, 5) and (9, -6)

• (5, 3) and (-6, 9)

• (5, 3) and (6, 9)

• (5, -3) and (-6, 9)

Multiple Choice

What is the slope of the line that passes through (4, 5) and (6, 6)?

• 2

• 1/2

• -2

• -1/2

x-intercepts are written (__, 0)

This means they are the x-values that make the equation equal to 0. Find them by setting the factors equal to 0 and solving.

Finding

x-intercepts​

Multiple Choice

x-intercepts are easiest to find from equations that are in factored form.

Which quadratic equation below is written in factored form?

• $$y=\left(x+6\right)\left(x-2\right)$$  

• $$y=\left(x+3\right)^2-9$$  

• $$y=-x^2+6x-4$$  

• $$y=\left(x-1\right)\left(x+1\right)+7$$  

Multiple Choice

In the equation $$y=\left(x-3\right)\left(x+8\right)\left(x-1\right)$$ , one x-intercept is $$x=3$$ . What are the other two x-intercepts?  

• 8 and 1

• -8 and -1

• 8 and -1

• -8 and 1

Multiple Choice

What are the x-intercepts of the equation $$y=\left(x+5.9\right)\left(x-2\right)$$ ?

• 5.9 and -2

• -5.9 and 2

• 5, 9, and -2

• 5, 9, and 2

Multiple Select

Select all relevant reasons that describe why $$x=12$$ is NOT an x-intercept for the function $$y=\left(x+12\right)\left(4x-12\right)$$    

• The factor $$\left(x+12\right)$$ corresponds to an x-intercept of -12, not 12.

• Since 12 appears twice in the equation, the x-intercept will be $$12^2$$  

• The +12 and -12 in the equation cancel out, so the only x-intercept is actually 0.

• The factor $$\left(4x-12\right)$$  does not equal 0 when x is 12. The corresponding x-intercept is 3.

Translations of functions are represented in function notation by adding or subtracting units either inside the parentheses (to move left/right) or outside the parentheses (to move up/down).

Translations of Functions

Multiple Choice

Which expression represents a translation of a function f to the right 1 unit?

• $$f\left(x-1\right)$$  

• $$f\left(x+1\right)$$  

• $$f\left(x\right)-1$$  

• $$f\left(x\right)+1$$  

Multiple Choice

Which expression represents a translation of a function k to the left 4 units and down 8 units?

• $$k\left(x+8\right)-4$$  

• $$k\left(x-8\right)-4$$  

• $$k\left(x+4\right)-8$$  

• $$k\left(x-4\right)-8$$  

Draw

Below is the graph of a function $$g\left(x\right)$$ . Sketch what $$g\left(x+1\right)+2$$   would look like.

Miscellaneous Topics

Multiple Choice

What is the sum of the two complex numbers $$4+3i$$  and $$7-i$$ ? 

• $$7+6i$$  

• $$11+2i$$  

• $$11+3i$$  

• $$28-3i^2$$  

Multiple Choice

For the function $$R\left(x\right)=\frac{12}{x}+1$$ , what is the value of $$R\left(6\right)-R\left(4\right)$$  ? 

• 3

• 4

• 1

• -1

Multiple Choice

Which of the following is equivalent to $$x^{\frac{1}{2}}$$ ? 

• $$\sqrt{x}$$  

• $$\frac{x}{2}$$  

• $$\frac{1}{x^2}$$  

• $$-x^2$$  

Multiple Choice

Which of the following is equivalent to $$m^{\frac{3}{8}}$$ ?

• $$\frac{3m}{8}$$  

• $$\sqrt[m]{3^8}$$  

• $$\sqrt[8]{m^3}$$  

• $$\sqrt[3]{m^8}$$