/*! 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 32 Find the horizontal and vertical... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 32

Find the horizontal and vertical intercepts of each equation. $$ k(x)=-5 x+1 $$

Short Answer

Expert verified
Horizontal intercept: \((\frac{1}{5}, 0)\), Vertical intercept: \((0, 1)\).

Step by step solution

01

Identify Intercepts

To find the intercepts, we need to determine the points where the function crosses the x-axis (horizontal intercept or x-intercept) and the y-axis (vertical intercept or y-intercept).
02

Find the Horizontal (x) Intercept

The horizontal intercept is found by setting the output of the function to zero and solving for the input. For \(k(x) = -5x + 1\), this means setting \(k(x) = 0\) and solving for \(x\).\[0 = -5x + 1 \5x = 1 \x = \frac{1}{5}\]Thus, the horizontal intercept is at \( (\frac{1}{5}, 0) \).
03

Find the Vertical (y) Intercept

The vertical intercept is found by evaluating the function at the input of zero. For \(k(x) = -5x + 1\), this means finding \(k(0)\).\[k(0) = -5(0) + 1 = 1\]Thus, the vertical intercept is at \( (0, 1) \).

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.

Horizontal Intercept
Imagine the horizontal intercept as the point where a line crosses the x-axis. This is the spot where the y-value of the function is zero. So, to find the horizontal intercept of a linear equation like \( k(x) = -5x + 1 \), we set the function equal to zero and solve for \( x \). This step is crucial because we're interested in pinpointing where the line meets the x-axis, making the y-value zero.

For instance:
  • Set the equation \( -5x + 1 \) to zero: \( 0 = -5x + 1 \).
  • Rearrange and solve for \( x \): \( 5x = 1 \).
  • Therefore, \( x = \frac{1}{5} \).
This gives you the horizontal intercept at \( (\frac{1}{5}, 0) \). The x-value is \(\frac{1}{5}\), while the y-value is zero, confirming its position on the x-axis.
Vertical Intercept
The vertical intercept is simply where the line touches the y-axis. At this junction, the x-value is zero. This makes it very easy to find because all you need to do is substitute \( x = 0 \) into the linear equation.

Here's how it works:
  • Take the equation \( k(x) = -5x + 1 \) and replace \( x \) with \( 0 \): \( k(0) = -5(0) + 1 \).
  • Calculate to find the y-value: \( k(0) = 1 \).
Now we know that the vertical intercept is \( (0, 1) \). Since the x-value is zero, it conveniently marks the point on the y-axis.
Linear Equation
A linear equation forms the backbone of straight-line graphs. In its simplest form, a linear equation is expressed as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This equation represents a straight line when plotted on a graph.

Key features of linear equations:
  • Constant rate of change: The slope, \( m \), indicates how steep the line is. It's a constant rate showing how much y increases or decreases as x changes.
  • Flat graph: If the slope is zero, the line is horizontal, indicating no change in y as x changes.
  • Simplicity: These equations only have variables that are raised to the power of one.
In summary, linear equations are straightforward, making them easy to plot and analyze. They provide a perfect foundation for recognizing patterns and predicting outputs given specific inputs.
Slope-Intercept Form
The slope-intercept form of a linear equation is a popular way to express lines due to its clarity and simplicity. The general form is \( y = mx + b \), where \( m \) represents the slope and \( b \) the y-intercept. Let's dive deeper:

  • The slope or \( m \) indicates the line's steepness. It tells us how y changes for each unit increase in x. A positive \( m \) means the line rises, while a negative one means it falls.
  • The y-intercept or \( b \) is the point where the line crosses the y-axis. It's the value of y when \( x \) is zero.
Understanding the slope-intercept form is crucial for quickly graphing linear equations and identifying their key properties. It allows you to easily spot changes in patterns and anticipate where the line will be on a graph.

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

Determine if each function is increasing or decreasing. $$ p(x)=\frac{1}{4} x-5 $$

Sketch each equation. $$ x=3 $$

Sketch each equation. $$ r(x)=4 $$

Sonya is currently 10 miles from home, and is walking further away at 2 miles per hour. Write an equation for her distance from home \(t\) hours from now.

Use algebra to find the point at which the line \(f(x)=-\frac{4}{5} x+\frac{274}{25}\) intersects the line \(h(x)=\frac{9}{4} x+\frac{73}{10}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.