### All SAT Mathematics Resources

## Example Questions

### Example Question #1 : Describing Exponential Vs. Linear Change

The following equation represents the change of a company’s market value since 1993: . Which of the following statements best describes this function?

**Possible Answers:**

This company is linearly growing by

This company is exponentially shrinking by

This company is linearly shrinking by

This company is exponentially growing by

**Correct answer:**

This company is exponentially shrinking by

The correct answer is “This company is exponentially shrinking by .” The above function is an exponential function of t because is an exponent of a base in the function. A linear equation would utilize as a coefficient, not an exponent. The best way to differentiate between growth and shrinkage in an exponential function is to see if the exponent’s base is greater or less than . If the base is , there would be no change in regardless of the value of , but if the base is less than , the company’s value is decreasing. so a loss of of market value annually.

### Example Question #1 : Word Translations & Equation Interpretation

The above table shows the growth of two plants in cm. Which of the following statements best describes the growth of these two plants?

**Possible Answers:**

Both Plant A and B show linear growth

Plant A shows exponential growth while Plant B is linear.

Both Plant A and B show exponential growth

Plant A shows linear growth while Plant B is exponential.

**Correct answer:**

Plant A shows exponential growth while Plant B is linear.

The correct answer is “Plant A shows exponential growth while Plant B is linear.” Each hour doubles the height of Plant A. Doubling is a characteristic of exponential growth . Each hour adds cm to the height of Plant B. Additive and subtractive trends are characteristic of linear growth .

### Example Question #1 : Describing Exponential Vs. Linear Change

Which of the following equations most closely describes the above graph?

**Possible Answers:**

**Correct answer:**

The correct answer is . The graph is curved and does not have a defined slope, so it cannot be a linear function . Now, between our exponential functions, we can see that at , , so this must be a coefficient outside of the exponential base. This only leaves . It might be one’s first thought to see that at , , pointing to the choice, but this equation does not work at any other value of .

### Example Question #2 : Describing Exponential Vs. Linear Change

Which of the following sets of equations accurately describes the above graph?

**Possible Answers:**

Option 2

Option 3

Option 4

Option 1

**Correct answer:**

Option 4

Both exponential equations should represent curved lines while the linear equations should be straight lines. shows exponential growth, so as increases, should increase. shows exponential decay, so as increases, should decrease. This narrows us down to Option 2 and 4. Within the linear equations, should be a steeper line than since the slope is greater, thus leaving us with Option 4 as the correct answer.

### Example Question #5 : Describing Exponential Vs. Linear Change

The population of moths in a given forest has been decreasing by every years since 2004. The population at the beginning of 2004 was . If represents the total moth population at time years after 2004, which following equations most closely describes the total moth population at any given time?

**Possible Answers:**

**Correct answer:**

Since the total moth population is a percentage of the previous period’s population, we know this is an exponential function and not a linear one. We also know that the population has been decreasing. An exponent base of would represent no change in the population, while a base of less than one would represent a decrease, so this rules out the option with as the exponent base. The population has been decreasing by so we can say the original of the population loses , leaving us with of the original population for every years that pass. Also, we use because the population only decreases by every years. This leaves us with .

### Example Question #1 : Describing Exponential Vs. Linear Change

The following equation represents the growth of bacteria in a petri dish: . Which of the following statements best describes this function?

**Possible Answers:**

This relationship is linear and the population of bacteria grows by bacteria every hour.

This relationship is exponential and doubles every hour.

This relationship is linear and the population of bacteria grows by bacteria every hour.

This relationship is exponential and grows times larger every hour.

**Correct answer:**

This relationship is exponential and doubles every hour.

The correct answer is “This relationship is exponential and doubles every hour.” Since the time variable is an exponent, we can narrow down our options and definitively state the function is exponential. The is an exponent with a base of , so as grows, the number of times is multiplied by itself also grows. Each multiplication of two is a doubling of the previous hour’s population (when , vs. when , ).

### Example Question #1 : Describing Exponential Vs. Linear Change

Which of the following is true about the following function in regards to time in hours: ?

I) This is a linear function

II) When , .

III) After hours, .

**Possible Answers:**

I and II

I

II

II and III

**Correct answer:**

II

The correct answer is II. This is an exponential function, not a linear function. When we plug in for , . When we plug in for , , not .

### Example Question #8 : Describing Exponential Vs. Linear Change

Maria has a fruit farm and wants to measure her apple and peach harvest from 2000 to 2020. Her apple harvest grew by approximately bushels per year while her peach harvest grew by bushels every years. In 2003, her apple yield was bushels while her peach harvest was bushels. What is the best estimate for the difference between apple and peach harvests in 2018?

**Possible Answers:**

**Correct answer:**

The correct answer is . Maria’s apple harvest follows an exponential pattern while her peach harvest is linear. Since time is relative to any given start point, we can call 2003 the year where . Her apple harvest *grows* by so when we convert our percent back to a decimal, we get an exponent base of . Recall, that if her harvest was *losing* of produce yearly, we would use . Since her starting harvest in 2003 was bushels, we can model her total bushel count as a function of time through the following equation: .

Her peach harvest is a linear function that grows by bushels every years. Since time is measured in single years and our growth is given in periods of years, be sure to take this into account in your linear equation . Since her starting harvest in 2003 was bushels, we can model her total bushel count as a function of time through the following equation: .

When we plug into our equation from subtracting 2003 from 2018, we get bushels of apples and bushels of peaches. .

### Example Question #9 : Describing Exponential Vs. Linear Change

Which of the following statements are true about an exponential function?

I) It takes the form

II) It changes at a constant rate per unit interval

III) It changes by a common ratio over equal intervals

**Possible Answers:**

II

I and III

I

I and II

**Correct answer:**

I and III

Exponential functions are in the form while linear are . Linear functions change at a constant rate per unit interval while exponential functions change by a common ratio over equal intervals.

### Example Question #10 : Describing Exponential Vs. Linear Change

Which of the following statements are true about the exponential function: ?

I) The y-intercept of this graph is

II) The base in this equation is

III) The x-intercept of this equation is

**Possible Answers:**

I

I and II

I and III

II

**Correct answer:**

I

When (the y-intercept), . The base of this equation is . Exponential functions never have x-intercepts unless they are in the form .

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