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Chapter 2: Problem 52

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ y=-3 $$

Short Answer

Expert verified
y-intercept is (0, -3); no x-intercept; horizontal line at y = -3.

Step by step solution

01

- Identify the Equation

Recognize the given equation: \[ y = -3 \]This is a horizontal line because the equation is in the form of a constant value for y.
02

- Find the y-intercept

For the y-intercept, set \(x = 0\) and solve for \(y\).Since the equation is already in the form \(y = -3\), the y-intercept is directly \(-3\).So, the y-intercept is (0, -3).
03

- Find the x-intercept

For the x-intercept, set \(y = 0\) and solve for \(x\).However, since \(y\) always equals \(-3\) and never 0, there is no x-intercept for this line.
04

- Graph the Equation

To graph the equation \(y = -3\), draw a horizontal line that crosses the y-axis at \(y = -3\). This line will be parallel to the x-axis.

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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 of a graph is where the line crosses the y-axis. This occurs when the value of x is zero. It's an important concept because it gives you a starting point for drawing the graph. For the equation y = -3y = -3, you can see that there's no x term, so the y-value is always -3, no matter what x is. Therefore, the y-intercept is at the point (0, -3).To find it, simply set x to 0 and solve for y. Here, since y is already -3 in the equation, the y-intercept is quite straightforward.
The concept of x-intercept
The x-intercept of a graph is where the line crosses the x-axis. This occurs when the value of y is zero. For most lines, you'll set y to 0 and solve for x to find the x-intercept.But in the case of the equation y = -3y = -3, you notice that the value of y is never zero 鈥� it is always -3. That's why this particular line does not have an x-intercept. Simply put, there's no point where this line crosses the x-axis.
Drawing a horizontal line
A horizontal line is a straight line that runs left to right and has the same y-value for every point along it. These lines are parallel to the x-axis. In the case of the equation y = -3y = -3, the line is completely horizontal because y is always -3 regardless of the value of x.To graph it, find the y-intercept at (0, -3) and draw a straight, horizontal line across the entire graph. This line will lie at the y-value of -3 continuously, never rising or falling.

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Most popular questions from this chapter

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following $$ f(a+b) $$

The table represents a linear function. (a) What is \(f(2)\) ? (b) If \(f(x)=-1.3,\) what is the value of \(x ?\) (c) What is the slope of the line? (d) What is the \(y\) -intercept of the line? (e) Using the answers from parts (c) and (d), write an equation for \(f(x)\). $$ \begin{array}{c|c} x & y=f(x) \\ \hline 0 & 3.5 \\ \hline 1 & 2.3 \\ \hline 2 & 1.1 \\ \hline 3 & -0.1 \\ \hline 4 & -1.3 \end{array} $$

Determine whether each pair of lines is parallel, perpendicular, or neither. \(x=6\) and \(6-x=8\)

Graph the intersection of each pair of inequalities. $$ 2 x-y \geq 2 \text { and } y<4 $$

Use your knowledge of the slopes of parallel and perpendicular lines. Is the figure with vertices at \((-11,-5),(-2,-19),(12,-10),\) and (3,4) a parallelogram? Is it a rectangle? (Hint: A rectangle is a parallelogram with a right angle.)
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