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Chapter 2: Problem 29

Find the horizontal and vertical intercepts of each equation. $$ f(x)=-x+2 $$

Short Answer

Expert verified
The vertical intercept is (0,2) and the horizontal intercept is (2,0).

Step by step solution

01

Understand the Equation

The given function is a linear equation: \[ f(x) = -x + 2 \]This equation is in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept (vertical intercept).
02

Find the Vertical Intercept

To find the vertical intercept, substitute \( x = 0 \) into the equation:\[ f(x) = -0 + 2 = 2 \]So the vertical intercept is \[ (0,2) \]
03

Find the Horizontal Intercept

The horizontal intercept occurs where \( y = 0 \), so set the equation to 0 and solve for \( x \):\[ 0 = -x + 2 \]\[ x = 2 \]Thus, the horizontal intercept is \[ (2,0) \]

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Key Concepts

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

Linear Equations
A linear equation is an equation that makes a straight line when it's graphed. These are the simplest kind of algebraic equations because they can be symbolized and graphed using only two variables, typically denoted as \( x \) and \( y \). In general, a linear equation can be written in the form \( ax + by = c \) where \( a \), \( b \), and \( c \) are constants. Linear equations are fundamental in mathematics because they describe one-to-one relationships and straightforward patterns. They are powerful tools for finding unknown values and describing linear progressions in the real world.
Slope-Intercept Form
The slope-intercept form is a way of writing linear equations. This form is particularly handy because it easily reveals both the slope of the line and its intercepts. The slope-intercept form is written as \( y = mx + b \), where:
  • \( m \) is the slope, which tells us the steepness of the line and the direction it's going.
  • \( b \) is the y-intercept (vertical intercept), the point where the line crosses the y-axis.
This format is extensively used to quickly graph linear equations and analyze their behavior. When you have a linear equation in this form, you can immediately identify the slope and the point where it cuts the y-axis.
The given equation \( f(x) = -x + 2 \) is already in slope-intercept form, where the slope \( m = -1 \) and the y-intercept \( b = 2 \).
Vertical Intercept
Finding the vertical intercept, also known as the y-intercept, involves determining the point where the line intersects the y-axis. By definition, this is where \( x = 0 \). To find this intercept, you substitute \( x = 0 \) into the equation and solve for \( y \).
Using the equation \( f(x) = -x + 2 \), substituting \( x = 0 \) gives us \( f(0) = -0 + 2 = 2 \). Therefore, the vertical intercept is at the point \( (0, 2) \). This means that when the line crosses the y-axis at \( y = 2 \). Vertical intercepts are essential for understanding where a line will start when graphing.
Horizontal Intercept
The horizontal intercept, also called the x-intercept, is the point where a line crosses the x-axis. To find it, you set the equation equal to zero (( y = 0 )) and solve for \( x \). This is because along the x-axis, the value of \( y \) is always zero.
For the equation \( f(x) = -x + 2 \), set \( f(x) = 0 \):
\[ 0 = -x + 2 \]
Solving for \( x \) gives \( x = 2 \). Hence, the horizontal intercept is at \( (2, 0) \). This step is crucial as it shows where the line crosses the x-axis, which can help in drawing the graph and understanding the solution's behavior across the axes.

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