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Chapter 2: Problem 20

Determine the \(y\) -intercept. $$ f(x)=-1.2 x+\frac{1}{5} $$

Short Answer

Expert verified
The y-intercept is \( \frac{1}{5} \).

Step by step solution

01

Understand the Concept of y-Intercept

The y-intercept of a function is the point where the graph of the function crosses the y-axis. This occurs when the value of x is zero. Therefore, to find the y-intercept, set x equal to zero and solve for the y-value.
02

Substitute x with 0 in the Function

Given the function \[ f(x) = -1.2x + \frac{1}{5} \] substitute \( x = 0 \): \[ f(0) = -1.2(0) + \frac{1}{5} \]
03

Simplify the Equation

Simplify the equation from Step 2: \[ f(0) = 0 + \frac{1}{5} \] \[ f(0) = \frac{1}{5} \]
04

State the y-Intercept

The y-intercept is the value of the function when \( x = 0 \). From the previous step, \( f(0) = \frac{1}{5} \). Therefore, the y-intercept is \( \frac{1}{5} \).

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Key Concepts

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

linear function
A linear function is a type of function that creates a straight line when graphed. It has the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope represents how steep the line is, and the y-intercept is where the line crosses the y-axis. In this exercise, our function is \( f(x) = -1.2x + \frac{1}{5} \). Here, the slope \( m \) is -1.2, and the y-intercept \( b \) is \( \frac{1}{5} \). This tells us that the line slopes downward and crosses the y-axis at \( 0.2 \). By understanding linear functions, we can predict and understand the behavior of simple real-world relationships.
graph of a function
The graph of a function shows all the points that satisfy the function. For a linear function like \( f(x) = -1.2x + \frac{1}{5} \), the graph is a straight line. We use the slope and y-intercept to draw this line.
  • Y-intercept: The starting point on the y-axis. In this function, it's \( \frac{1}{5} \).
  • Slope: The rate at which the line rises or falls as x increases. Here, the slope is -1.2, indicating the line descends as we move from left to right.
To graph it, you start at the y-intercept \( (0, \frac{1}{5}) \), then use the slope to find other points. For example, from \( (0, \frac{1}{5}) \), if x increases by 1, y decreases by 1.2, positioning us at \( (1, -1) \). Plot a few more points and connect them for the complete line.
solving equations
Solving equations helps us find unknown values. With linear functions, one of the typical tasks is finding intercepts. To find the y-intercept, we set x to 0. For the equation \( f(x) = -1.2x + \frac{1}{5} \):
  • Substitute x with 0: \( f(0) = -1.2(0) + \frac{1}{5} \)
  • Simplify: \( f(0) = 0 + \frac{1}{5} \)
  • Result: \( f(0) = \frac{1}{5} \)
This calculation shows the y-intercept is \( \frac{1}{5} \), meaning the graph of the function crosses the y-axis at 0.2. By practicing these steps, you can solve various linear function problems with ease and confidence.

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Most popular questions from this chapter

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