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Chapter 0: Problem 12

Determine the intercepts of the graphs of the following equations. $$f(x)=-\frac{1}{2} x-1$$

Short Answer

Expert verified
The y-intercept is (0, -1) and the x-intercept is (-2, 0).

Step by step solution

01

Identify the y-intercept

The y-intercept of a function is the point where the graph crosses the y-axis. This occurs when the x-coordinate is 0. To find the y-intercept of the function \( f(x) = -\frac{1}{2} x - 1 \), substitute 0 for x in the equation.
02

Calculate the y-intercept

Substitute \( x = 0 \) into the equation: \[ f(0) = -\frac{1}{2}(0) - 1 = -1 \] The y-intercept is the point \( (0, -1) \)
03

Identify the x-intercept

The x-intercept of a function is the point where the graph crosses the x-axis. This occurs when the y-coordinate (or \( f(x) \)) is 0. To find the x-intercept, set \( f(x) \) to 0 in the equation and solve for x.
04

Calculate the x-intercept

Set \( f(x) = 0 \) and solve for x: \[ 0 = -\frac{1}{2} x - 1 \] Add 1 to both sides: \[ 1 = -\frac{1}{2} x \] Multiply both sides by -2 to solve for x: \[ x = -2 \] The x-intercept is the point \( (-2, 0) \)

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Key Concepts

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

Understanding the Y-Intercept
The y-intercept is where the graph of a function crosses the y-axis. This happens when the x-coordinate is zero.
For the function given, \(f(x) = -\frac{1}{2} x - 1\), we find the y-intercept by setting \(x = 0\).
  • We substitute zero into the equation: \ \(f(0) = -\frac{1}{2} \cdot 0 - 1 = -1\).
So, the y-intercept of this function is the point \((0, -1)\).
This is crucial because it tells us where the graph starts when plotting in a coordinate system.
Being able to find the y-intercept helps in understanding the function's behavior and graphing it accurately.
Finding the X-Intercept
The x-intercept is the point where the graph crosses the x-axis. This point occurs when the y-coordinate (the value of the function \(f(x)\)) is zero.
For the function \(f(x)=-\frac{1}{2} x - 1\), we find the x-intercept by setting \(f(x)\) to zero and solving for \(x\).
  • We set up the equation: \ \(0 = -\frac{1}{2} x - 1\).
  • Add 1 to both sides to isolate \(x\)-term: \ \(0 + 1 = -\frac{1}{2} x - 1 + 1\) \ \(1 = -\frac{1}{2} x\).
  • Multiply both sides by -2 to solve for \(x\): \ \(1 \cdot -2 = -\frac{1}{2} x \cdot -2\) \ \(x = -2\).
Thus, the x-intercept is \(( -2, 0 )\).
This tells us the location on the graph where the function value is zero, which is very useful for graphing and understanding the function鈥檚 behavior.
Solving Linear Equations to Find Intercepts
To find the intercepts of a linear function, you need to solve simple linear equations.
The principles behind solving these equations are:
  • Substitution: For the y-intercept, substitute \(x = 0\) in \(f(x)\).
  • Setting the function to zero: For the x-intercept, set \(f(x) = 0\) and solve for \(x\).
For example, in \(f(x) = -\frac{1}{2} x - 1\):
  • For the y-intercept: \ \begin{align*} f(0) = -\frac{1}{2} \cdot 0 - 1 = -1 \ \text{The y-intercept is } (0, -1). \ \text{Solution: Substitute } x = 0.
  • For the x-intercept: \ \begin{align*} 0 = -\frac{1}{2} x - 1 \ 1 = -\frac{1}{2} x \ \text{Solve: Multiply by -2:} \ x = -2, \ \text{Thus, the x-intercept is } (-2, 0).
Learning to solve these equations efficiently is key to mastering the concept of intercepts in linear functions.
This skill will help you graph and analyze linear equations accurately.

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