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Chapter 2: Problem 4

Sketch the graph of the equation, and label the \(x\) - and \(y\) -intercepts. $$y=-2 x-3$$

Short Answer

Expert verified
The graph is a line crossing the y-axis at (0, -3) and the x-axis at (-3/2, 0).

Step by step solution

01

Identify the Equation Type

The equation given is in the slope-intercept form, which is written as \(y = mx + b\). In the equation \(y = -2x - 3\), the slope \(m\) is -2, and the y-intercept \(b\) is -3.
02

Find the y-intercept

The y-intercept is the value of \(y\) when \(x = 0\). Substitute \(x = 0\) into the equation: \(y = -2(0) - 3\). This simplifies to \(y = -3\). Therefore, the y-intercept is the point \((0, -3)\).
03

Find the x-intercept

The x-intercept is the value of \(x\) when \(y = 0\). Set \(y = 0\) in the equation: \(0 = -2x - 3\). Solve for \(x\): \(2x = -3\), so \(x = -\frac{3}{2}\). The x-intercept is the point \((-\frac{3}{2}, 0)\).
04

Sketch the Graph

Plot the intercepts found in Step 2 and Step 3: \((0, -3)\) for the y-intercept and \((-\frac{3}{2}, 0)\) for the x-intercept. Draw a line through these two points, extending in both directions to form the graph of the line \(y = -2x - 3\).
05

Verify the Graph

Check the graph with the equation鈥檚 slope. The slope \(-2\) indicates the line drops 2 units vertically for each unit it moves 1 unit horizontally to the right. Confirm this relationship between points on the graph to ensure the sketch is accurate.

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Key Concepts

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

Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most commonly used formats in algebra. It is expressed as \( y = mx + b \). In this structure:
  • \( m \) represents the slope of the line, indicating how steep the line is.
  • \( b \) stands for the y-intercept, which is the point where the line crosses the y-axis.
Applying this to the equation \( y = -2x - 3 \), we identify that:
  • The slope \( m \) is \( -2 \). This means for every 1 unit you move to the right (positive direction on x-axis), the line goes down 2 units on the y-axis.
  • The y-intercept \( b \) is \( -3 \). Thus, the line crosses the y-axis at the point \((0, -3)\).
The slope-intercept form simplifies the process of graphing a linear equation and analyzing how different components affect the line. It's especially helpful to immediately see both the rate of change and a key point on the graph.
x-intercepts
The x-intercept of a line is where the graph crosses the x-axis. At this point, the value of \( y \) is zero. To find this intercept for an equation, simply set \( y = 0 \) and solve for \( x \).

In our example, the equation is \( y = -2x - 3 \). To find the x-intercept:
  • Set \( y = 0 \): \( 0 = -2x - 3 \).
  • Add 3 to both sides: \( 3 = -2x \).
  • Divide both sides by -2: \( x = -\frac{3}{2} \).
Thus, the x-intercept is \( (-\frac{3}{2}, 0) \).
Understanding x-intercepts is crucial when graphing because it lets us find a key point that helps in drawing the line accurately.
y-intercepts
The y-intercept occurs where a graph crosses the y-axis. At this coordinate, \( x = 0 \). Finding the y-intercept from an equation lets you quickly determine one point through which the line passes.

For the equation \( y = -2x - 3 \), you can calculate the y-intercept by:
  • Substituting \( x = 0 \) into the equation: \( y = -2(0) - 3 \).
  • This simplifies to \( y = -3 \).
So, the y-intercept is at the point \((0, -3)\).
Recognizing the y-intercept is fundamental in graphing linear equations. It's the starting point when plotting a line on a graph, providing a concrete place to begin before using the slope to determine other points.

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Most popular questions from this chapter

The symbol \([x]\) denotes values of the greatest integer function. Sketch the graph of \(f\) (a) \(f(x)=[x-3]\) (b) \(f(x)=[x]-3\) (c) \(f(x)=2[x]\) (d) \(f(x)=[2 x]\) (e) \(f(x)=[-x]\)

(a) Use the quadratic formula to find the zeros of \(f .\) (b) Find the maximum or minimum value of \(f(x)\) (c) Sketch the graph of \(f\) $$f(x)=9 x^{2}+24 x+16$$

Find (a) ( \(f \circ g)(x)\) and the domain of \(f \circ g\) and (b) \((g \circ f)(x)\) and the domain of \(g \circ f\) $$f(x)=\frac{x-1}{x-2}, \quad \quad \quad \quad g(x)=\frac{x-3}{x-4}$$

Sketch the graph of the equation. $$y=|\sqrt{x}-2|$$

Find (a) ( \(f \circ g)(x)\) and the domain of \(f \circ g\) and (b) \((g \circ f)(x)\) and the domain of \(g \circ f\) $$f(x)=\frac{x}{x-2}, \quad g(x)=\frac{3}{x}$$
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