/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Graph each function using the sl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 39

Graph each function using the slope and \(y\) -intercept. $$h(x)=\frac{3}{5} x-2$$

Short Answer

Expert verified
The graph of \(h(x) = \frac{3}{5}x - 2\) has a slope of \(\frac{3}{5}\) and a y-intercept of -2. Plot the y-intercept (0, -2), then use the slope to find another point (5, 1). Connect these points with a straight line to complete the graph.

Step by step solution

01

Identify the slope and y-intercept

The given function is in the slope-intercept form (\(y = mx + b\)), where m is the slope and b is the y-intercept. In our case, \(h(x) = \frac{3}{5}x - 2\), the slope (m) is \(\frac{3}{5}\) and the y-intercept (b) is -2.
02

Plot the y-intercept

Start by plotting the y-intercept on the graph, which is the point (0, -2). This is where the line crosses the y-axis.
03

Use the slope to find another point

Recall that slope is the change in y divided by the change in x or \(\frac{螖y}{螖x}\). In our case, the slope is \(\frac{3}{5}\), meaning that for every 5 units we move to the right on the x-axis, we will move 3 units up on the y-axis. From the y-intercept (0, -2), count 5 units to the right and 3 units up, and you will reach the point (5, 1). Plot this point on the graph.
04

Draw the line

Connect the two points (0, -2) and (5, 1) with a straight line. This line represents the graph of the function \(h(x) = \frac{3}{5}x - 2\). The graph is now complete with the slope of \(\frac{3}{5}\) and y-intercept of -2.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Slope-Intercept Form
The slope-intercept form is a way of writing the equation of a straight line. It's typically expressed as \(y = mx + b\). Here, \(m\) denotes the slope of the line, and \(b\) is the y-intercept. This form is extremely handy for graphing because it directly gives us the slope and y-intercept.
When you have an equation such as \(y = \frac{3}{5}x - 2\), it鈥檚 easy to see that the slope \(m\) is \(\frac{3}{5}\) and the y-intercept \(b\) is -2.
  • The slope \(m\) tells you how steep the line is.
  • The y-intercept \(b\) shows where the line crosses the y-axis.
Identifying these values is the first essential step in graphing a linear function.
Slope Calculation
Slope is a measure of how much a line inclines or declines as you move along it. In the slope-intercept form equation, \(y = mx + b\), the slope \(m\) is represented as a fraction \(\frac{螖y}{螖x}\).
The fraction \(\frac{3}{5}\) means that for every 5 units you move horizontally to the right along the x-axis, you move 3 units vertically up along the y-axis. This concept is fundamental because it defines the gradient and direction of the line.
  • Slope \(\frac{3}{5}\): Rise 3 units, Run 5 units.
  • A positive slope indicates the line ascends as it moves from left to right.
By understanding slope calculation, you can determine how a line behaves just by looking at its equation.
Plotting Points
Plotting points is a vital step in creating the graph of a linear equation. You begin with the y-intercept point, which indicates where the line crosses the y-axis.
In our example equation \(y = \frac{3}{5}x - 2\), the y-intercept is at (0, -2). This is the first point you should plot on your graph.
Next, use the slope to find a second point. Starting from the y-intercept, move according to the slope, \(\frac{3}{5}\). You move 5 units to the right (x direction) and 3 units up (y direction) to find the second point, which is at (5, 1).
  • Plot the y-intercept: (0, -2)
  • Using the slope, find the second point: (5, 1)
Accurately plotting points ensures that your graph correctly represents the given function.
Graphing Techniques
Graphing linear functions involves connecting the plotted points to form a straight line. After determining the correct points, such as (0, -2) and (5, 1) in our example, draw a line through these points.
The line should extend indefinitely in both directions, as represented by small arrowheads at each end of the line. This denotes that the function continues beyond the drawn segment.
  • Ensure the line is straight by using a ruler tool if drawing by hand.
  • Extend the line with arrowheads to show continuity.
Employing these graphing techniques will produce an accurate representation of the function like \(h(x) = \frac{3}{5}x - 2\). Completing these steps allows for a clear visual understanding of the relationship between the variables x and y.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The velocity of an object, \(v\) (in feet per second), of an object during free- fall \(t\) sec after being dropped can be defined by the function $$ v(t)=32 t $$ a) Find the velocity of an object 1 sec after being dropped. b) Find the velocity of an object 3 sec after being dropped. c) When will the object be traveling at 256 ft / sec? d) Graph the function.

Write the slope-intercept form of the equation of the line, if possible, given the following information. \(m=4\) and contains \((-4,-1)\)

Write an equation of the line parallel to the given line and containing the given point. Write the answer in slope intercept form or in standard form, as indicated. $$x+4 y=32 ;(-8,5) ; \text { standard form }$$

Graph each function using the slope and \(y\) -intercept. $$g(x)=3 x+3$$

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. perpendicular to \(2 x-6 y=-3\) containing \((2,2)\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.