The function is graphed in the -plane for . A line passes through the two points on the graph of whose -coordinates are and . What is the slope of this line?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 117 | Free SAT Prep | Aniko

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Question 117·Hard·Nonlinear Functions

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The function $f(x)=\ln(3x-2)$ is graphed in the $xy$ -plane for $x>\tfrac23$ . A line passes through the two points on the graph of $f$ whose $x$ -coordinates are $1$ and $7$ . What is the slope of this line?

For questions asking for the slope of a line through two points on a function’s graph, immediately translate it to the average rate of change formula $\dfrac{f(b)-f(a)}{b-a}$ . Plug in the given $x$ -values to find the corresponding $f(x)$ values, simplify carefully (using key facts like $\text{ln}(1)=0$ and valid log rules), and only at the end divide by the difference in $x$ . Avoid trying to combine everything into one logarithm unless you are using a correct property, and always double-check that the denominator is the difference of the $x$ -coordinates, not any number taken from inside the function.

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Relate the question to slope between two points

Think of the two points on the graph of $f$ as $(1,f(1))$ and $(7,f(7))$ . What is the general formula for the slope between $(x_1,y_1)$ and $(x_2,y_2)$ ?

Find the function values at the given $x$ -coordinates

Compute $f(1)$ and $f(7)$ using $f(x)=\text{ln}(3x-2)$ . Be careful to simplify $3x-2$ first before taking the natural logarithm.

Use a key logarithm fact

Once you have $f(1)$ and $f(7)$ , remember the special value $\text{ln}(1)$ to simplify the numerator in the slope formula.

Finish the average rate of change

Substitute your values of $f(1)$ and $f(7)$ into $\dfrac{f(7)-f(1)}{7-1}$ and simplify the fraction.

Desmos Guide

Define the function in Desmos

In Desmos, type f(x) = ln(3x - 2) to match the given function.

Evaluate the function at the two x-values

In a new line, type f(1) and f(7) to see the $y$ -values at $x=1$ and $x=7$ . These are the coordinates of the two points on the graph.

Compute the slope as an average rate of change

In another line, type (f(7) - f(1)) / (7 - 1) to have Desmos calculate the slope between the two points. The numerical result that appears is the slope; match it to the correct algebraic expression in the choices.

Step-by-step Explanation

Use the slope (average rate of change) formula

The line passes through the two points on the graph where $x=1$ and $x=7$ .

Those points are $(1,f(1))$ and $(7,f(7))$ .

The slope $m$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is

m = \frac{y_2 - y_1}{x_2 - x_1}.

Here, that becomes

m = \frac{f(7) - f(1)}{7 - 1}.

Compute $f(1)$ and $f(7)$

The function is $f(x) = \text{ln}(3x - 2)$ .

For $x=1$ :

f(1) = \text{ln}(3\cdot 1 - 2) = \text{ln}(1).

For $x=7$ :

f(7) = \text{ln}(3\cdot 7 - 2) = \text{ln}(19).

Substitute these into the slope expression:

m = \frac{\text{ln}(19) - \text{ln}(1)}{7 - 1}.

Simplify the logarithms and the fraction

First, use the fact that $\text{ln}(1) = 0$ :

\text{ln}(19) - \text{ln}(1) = \text{ln}(19) - 0 = \text{ln}(19).

The denominator is $7 - 1 = 6$ , so

m = \frac{\text{ln}(19)}{6}.

So, the slope of the line is $\dfrac{\text{ln}19}{6}$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation