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Chapter 2: Problem 5

Decide whether each statement is true or false. If the statement is false, tell why. The graph of \(y=2 x+4\) has \(x\) -intercept \(-2\) and \(y\) -intercept 4

Short Answer

Expert verified
True, the intercepts given are correct.

Step by step solution

01

Identify the equation

The given equation is a linear equation in the slope-intercept form, which is represented as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. For this particular equation, \(y = 2x + 4\), the slope \(m\) is 2 and the y-intercept \(b\) is 4.
02

Find the y-intercept

To find the y-intercept, set \(x = 0\) in the equation and solve for \(y\). \(y = 2(0) + 4\) Thus, \(y = 4\). So, the y-intercept is 4, and the corresponding coordinate point is \((0, 4)\).
03

Find the x-intercept

To find the x-intercept, set \(y = 0\) in the equation and solve for \(x\). \(0 = 2x + 4\) Subtract 4 from both sides: \(-4 = 2x\) Divide by 2: \(x = -2\). Thus, the x-intercept is -2, and the corresponding coordinate point is \((-2, 0)\).
04

Evaluate the statement

Compare the calculated intercepts with the given statement. The statement says the graph of \(y = 2x + 4\) has an x-intercept of -2 and a y-intercept of 4. Both calculated intercepts match the statement.
05

Conclusion

Since both intercepts in the statement are correct, the statement is true.

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Key Concepts

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

slope-intercept form
Linear equations can often be represented using the slope-intercept form, which is written as \( y = mx + b \). In this format, \( m \) represents the slope of the line, and \( b \) is the y-intercept. Slope indicates how steep the line is, while the y-intercept is where the line crosses the y-axis. This form makes it easy to quickly identify significant features of the line.
x-intercept
The x-intercept of a line is the point where the line crosses the x-axis. To find the x-intercept, set \( y = 0 \) in the equation and solve for \( x \). For example, for the equation \( y = 2x + 4 \), set \( y = 0 \):
\( 0 = 2x + 4 \).
Subtract 4 from both sides: \( -4 = 2x \).
Finally, divide by 2: \( x = -2 \).
So, the x-intercept is \( -2 \), and it corresponds to the point \( (-2, 0) \).
y-intercept
The y-intercept is the point where the line crosses the y-axis. This is found by setting \( x = 0 \) in the equation and solving for \( y \). For the equation \( y = 2x + 4 \), set \( x = 0 \): \( y = 2(0) + 4 \). Simplifying this, we get: \( y = 4 \). So, the y-intercept is \( 4 \), corresponding to the point \( (0, 4) \).
graphing linear equations
Graphing linear equations involves plotting points that satisfy the equation and then connecting them to form a straight line. Here鈥檚 how you can graph \( y = 2x + 4 \):
  1. Identify and plot the y-intercept. In this case, it's \( (0, 4) \).
  2. Identify and plot the x-intercept. For this line, it's \( (-2, 0) \).
  3. Draw a straight line that passes through both points. This line represents all solutions to the equation \( y = 2x + 4 \).
This method helps visualize the relationship between variables and makes it easy to identify significant features of the line.

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Most popular questions from this chapter

Give the slope and y-intercept of each line, and graph it. $$x+2 y=-4$$

Let \(f(x)=x^{2}+3\) and \(g(x)=-2 x+6 .\) Find each of the following. $$(f-g)(4)$$

Write an equation (a) in standard form and (b) in slope-intercept form for the line described through \((-1,4),\) parallel to \(x+3 y=5\)

Let \(f(x)=x^{2}+3\) and \(g(x)=-2 x+6 .\) Find each of the following. $$(f+g)(-5)$$

For certain pairs of functions \(f\) and \(g,(f \circ g)(x)=x\) and \((g \circ f)(x)=x .\) Show that this is true for each pair. $$f(x)=4 x+2, g(x)=\frac{1}{4}(x-2)$$
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