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Chapter 3: Problem 4

Exer. 1-20: Sketch the graph of the equation, and label the \(x\) - and \(y\)-intercepts. $$ y=-2 x-3 $$

Short Answer

Expert verified
Y-intercept: (0, -3); X-intercept: \(-\frac{3}{2}, 0\).

Step by step solution

01

Understand the Form of the Equation

The given equation is in the form of a linear equation, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, the equation is \( y = -2x - 3 \).
02

Identify the Y-intercept

The y-intercept is the value of \( y \) when \( x \) is 0. By setting \( x = 0 \) in the equation, we have \( y = -3 \). Thus, the y-intercept is \( (0, -3) \).
03

Find the X-intercept

The x-intercept is the value of \( x \) when \( y \) is 0. Set \( y = 0 \) in the equation and solve for \( x \):\[ 0 = -2x - 3 \]Add 3 to both sides:\[ 3 = -2x \]Divide by -2:\[ x = -\frac{3}{2} \]Thus, the x-intercept is \( \left(-\frac{3}{2}, 0\right) \).
04

Sketch the Graph

Plot the x-intercept \( \left(-\frac{3}{2}, 0\right) \) and the y-intercept \( (0, -3) \) on the coordinate plane. Draw a straight line through these two points since the equation represents a linear function.
05

Review the Graph

Make sure the line has a downward slope, consistent with the negative slope \( m = -2 \). The line passes through the intercept point \( (0, -3) \) and \( \left(-\frac{3}{2}, 0\right) \), confirming correctness.

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Key Concepts

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

graphing linear equations
Graphing a linear equation involves creating a visual representation of the equation on a coordinate plane. Linear equations are represented by straight lines, and they typically follow the format:
  • \( y = mx + b \)
Here, \( m \) is the slope of the line, indicating how steep it is, while \( b \) is the y-intercept, the point where the line crosses the y-axis.To graph a linear equation such as \( y = -2x - 3 \), start by identifying key points like the x- and y-intercepts. Once these points are plotted, draw a straight line through them. This line visually represents all the possible solutions to the equation. Remember, since \( m = -2 \) is negative in this example, the slope of the line should be downward from left to right.
x-intercepts
X-intercepts are where the graph of the equation crosses the x-axis. At these points, the value of \( y \) is zero. To find the x-intercept for \( y = -2x - 3 \), replace \( y \) with 0 and solve for \( x \):
Solve \( 0 = -2x - 3 \), by adding 3 to both sides:\[ 3 = -2x \]Then divide by -2:\[ x = -\frac{3}{2} \]This calculation shows that the x-intercept is \( \left(-\frac{3}{2}, 0\right) \). Knowing how to find the x-intercept is crucial for graphing because it provides a critical point through which the line will pass.
y-intercepts
The y-intercept is the point at which the graph crosses the y-axis. At the y-intercept, the value of \( x \) is zero. To determine the y-intercept of \( y = -2x - 3 \), substitute \( x = 0 \) into the equation:Substitute as follows:\[ y = -2(0) - 3 \]\[ y = -3 \]This tells us that the y-intercept is \( (0, -3) \). Plotting the y-intercept on the graph is a great start to visualizing the linear equation. Like x-intercepts, y-intercepts help define the trajectory of the graphed line and are essential for accurately drawing linear relationships.

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Most popular questions from this chapter

Exer. 3-12: Determine whether \(f\) is even, odd, or neither even nor odd. $$ f(x)=12 $$

If \(f\) is an odd function and \(g\) is an even function, is \(f g\) even, odd, or neither even nor odd?

Exer. 27-32: If the point \(P\) is on the graph of a function \(f\), find the corresponding point on the graph of the given function. $$ P(3,-2) ; \quad y=2 f(x-4)+1 $$

Exer. 47-52: Sketch the graph of \(f\). $$ f(x)= \begin{cases}-2 x & \text { if } x<-1 \\ x^{2} & \text { if }-1 \leq x<1 \\ -2 & \text { if } x \geq 1\end{cases} $$

Exer. 3-12: Determine whether \(f\) is even, odd, or neither even nor odd. $$ f(x)=\sqrt[3]{x^{3}-x} $$
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