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Chapter 2: Problem 90

Write a rule for a linear function \(y=k(x),\) given that \(k(-2)=10\) and \(k(5)=-18\).

Short Answer

Expert verified
The linear function is \(y = -4x + 2\).

Step by step solution

01

Identify Points

The function is given by two points: (-2, 10) and (5, -18). These points will help determine the slope (m) and the y-intercept (b) of the linear function.
02

Calculate the Slope

Use the slope formula \[ m=\frac{y_2 - y_1}{x_2 - x_1} \]to find the slope of the function. In this case, \[ m=\frac{-18 - 10}{5 - (-2)}=\frac{-28}{7}=-4\].
03

Write the Slope-Intercept Form

A linear function can be expressed in the form \[ y = mx + b \]. At this stage, substitute \( m = -4 \) and use one of the points to find \( b \).
04

Solve for y-intercept

Using the point (-2, 10):\[ 10 = -4(-2) + b \]. Simplify to get \[ 10 = 8 + b \] then \[ b = 2 \].
05

Formulate the Function

Substitute the slope (m) and y-intercept (b) into the slope-intercept form: \[ y = -4x + 2 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
To understand linear functions, it's essential to grasp the slope-intercept form. This is a simple and common way to write a linear equation. The general form is (\[ y = mx + b \]). Here:
  • \( y \) indicates the dependent variable (the output of the function).
  • \( x \) is the independent variable (the input).
  • \( m \) represents the slope of the line, showing how steep it is.
  • \( b \) is the y-intercept, where the line crosses the y-axis.
This form makes it easy to plot a linear function on a graph and analyze the relationship between \( x \) and \( y \). For instance, in the given problem, we find the final equation as \( y = -4x + 2 \), which is in slope-intercept form.
calculating slope
The slope of a line (\( m \)) measures its steepness and direction. To calculate the slope between two points, use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of the two points.
Let's break it down:
  • Subtract the y-values of the points: \( y_2 - y_1 \)
  • Subtract the x-values of the points: \( x_2 - x_1 \)
  • Divide the difference in y-values by the difference in x-values.
In the problem, for points (-2, 10) and (5, -18):
\[ m = \frac{-18 - 10}{5 - (-2)} = \frac{-28}{7} = -4 \]
The slope \( m \) is -4, meaning the line falls 4 units for every 1 unit it moves to the right.
y-intercept
The y-intercept (\( b \)) is the point where the line crosses the y-axis. This occurs when \( x = 0 \). To find the y-intercept, we use the slope-intercept formula \( y = mx + b \) and substitute one of our points and the slope into it.
From the example, we use the point (-2, 10) and the calculated slope (-4):
\[ 10 = -4(-2) + b \]
Simplifying:
\[ 10 = 8 + b \]
Subtract 8 from both sides to solve for \( b \):
\[ b = 2 \]
This means our line intersects the y-axis at (0, 2).
Combining the slope and y-intercept, we get the equation \( y = -4x + 2 \) as our final linear function.

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Most popular questions from this chapter

Given \(f(x)=\sqrt{x-4},\) a. Find the difference quotient. b. Rationalize the numerator of the expression in part (a) and simplify. c. Evaluate the expression in part (b) for \(h=0\).

Given \(f(x)=\sqrt{x+3}\) a. Find the difference quotient. b. Rationalize the numerator of the expression in part (a) and simplify. c. Evaluate the expression in part (b) for \(h=0\).

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=5 x+9$$

Explain what it means for a function to be increasing on an interval.

A website designer creates videos on how to create websites. She sells the videos in 10 -hr packages for \(\$ 40\) each. Her one-time initial cost to produce each 10 -hr video package is \(\$ 5000\) (this includes labor and the cost of computer supplies). The cost to package and ship each \(\mathrm{CD}\) is \(\$ 2.80\). a. Write a linear cost function that represents the \(\operatorname{cost} C(x)\) to produce, package, and ship \(x\) 10-hr video packages. b. Write a linear revenue function to represent the revenue \(R(x)\) for selling \(x\) 10-hr video packages. c. Evaluate \((R-C)(x)\) and interpret its meaning in the context of this problem. d. Determine the profit if the website designer produces and sells 2400 video packages in the course of one year.
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