Question

$$ f(x)=5 x-2 $$ (a) Find the average rate of change from 1 to 3 . (b) Find an equation of the secant line containing \((1, f(1))\) and \((3, f(3))\)

Short answer

(a) 5, (b) \( y = 5x - 2 \)

Step-by-step solution

Evaluate the function at given points

First, find the values of the function at the points 1 and 3.Calculate \( f(1) \):\[ f(1) = 5(1) - 2 = 5 - 2 = 3 \]Calculate \( f(3) \):\[ f(3) = 5(3) - 2 = 15 - 2 = 13 \]

Calculate the average rate of change

The average rate of change of a function between two points \( x_1 \) and \( x_2 \) is given by the formula:\[ \text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]In this case, \( x_1 = 1 \) and \( x_2 = 3 \). So, we have:\[ \text{Average rate of change} = \frac{f(3) - f(1)}{3 - 1} = \frac{13 - 3}{3 - 1} = \frac{10}{2} = 5 \]

Find the slope for the secant line

The slope of the secant line is the same as the average rate of change calculated in Step 2.So, the slope \( m \) of the secant line is 5.

Use point-slope form to find the equation of the secant line

The point-slope form of a line's equation is given by:\[ y - y_1 = m(x - x_1) \]Here, one of the points is \((1, f(1)) = (1, 3)\) and the slope \( m \) is 5. Substitute the values into the equation:\[ y - 3 = 5(x - 1) \]Simplify to find the equation in slope-intercept form:\[ y - 3 = 5x - 5 \]\[ y = 5x - 2 \]

Key concepts

linear functions

A linear function is one of the simplest types of functions in algebra. It's often written in the form of \( f(x) = mx + b \), where 'm' represents the slope and 'b' is the y-intercept. Linear functions produce a straight line when graphed. This makes them incredibly useful for modeling relationships where the change is consistent.
For example, in the original exercise, the function provided is \( f(x) = 5x - 2 \). Here, '5' is the slope of the line, meaning for every unit increase in 'x', 'f(x)' increases by 5. The '-2' is the y-intercept, which is the point where the line crosses the y-axis.
Understanding linear functions is crucial for topics like the average rate of change and the slope of a secant line. They form the foundation for more complex mathematical concepts.

slope of a secant line

The slope of a secant line is key to understanding how changes occur over an interval. When dealing with a curve, a secant line intersects the curve at two points. The slope of this secant line gives us an average rate of change between these two points.
In the provided exercise, the secant line passes through the points \( (1, f(1)) \) and \( (3, f(3)) \). We calculated \( f(1) = 3 \) and \( f(3) = 13 \).
Using the formula for the average rate of change, which is basically the same as the slope of the secant line, we get: \[ \text{{Average rate of change}} = \frac{{f(x_2) - f(x_1)}}{{x_2 - x_1}} = \frac{{13 - 3}}{{3 - 1}} = \frac{{10}}{{2}} = 5 \] This '5' is the slope of the secant line. Understanding this slope helps us further analyze the behavior of the function over that interval.

point-slope form

The point-slope form is a straightforward way to write the equation of a line when you know the slope and one point on the line. The general form is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the given point.
In the exercise, we found that the slope \( m \) is 5, and we used the point \( (1, 3) \). Substituting these values, the point-slope form becomes: \[ y - 3 = 5(x - 1) \] Simplifying this, you get: \[ y - 3 = 5x - 5 \] \[ y = 5x - 2 \] This equation is now in slope-intercept form. Using the point-slope form makes it easier to determine the equation of a line quickly and accurately, especially when you already know the slope and a point through which the line passes.