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Chapter 3: Problem 46

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

Short Answer

Expert verified
x-intercept: (-3, 0); y-intercept: none.

Step by step solution

01

Identify the Equation

The given equation is a vertical line, represented by the equation \(x = -3\).
02

Find the x-Intercept

Since the line is vertical and intersects the x-axis at \(x = -3\), the x-intercept is \((-3, 0)\).
03

Find the y-Intercept

Vertical lines do not cross the y-axis, so there is no y-intercept.
04

Graph the Equation

Draw a vertical line through the point \((-3, 0)\). This represents the graph of the equation \(x = -3\).

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Key Concepts

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

x-intercept
The concept of the x-intercept is fundamental when graphing linear equations. The x-intercept is the point where the graph of a line crosses the x-axis.
At this point, the value of the function (y-coordinate) is zero. To find the x-intercept of a line represented by an equation, set the y-value to 0 and solve for x.
Let's look at our example: the equation is given as \(x = -3\).
Since the value of x is given as a constant, \(x = -3\), this equation tells us that the line will intersect the x-axis at the point \((-3, 0)\).
Remember, for other types of equations, setting y to 0 will allow you to solve for the x-intercept.
y-intercept
The y-intercept refers to the point where a line crosses the y-axis.
At this point, the value of x is zero. To find the y-intercept of a line, set the x-value to 0 and solve for y.
For our specific equation, \(x = -3\), we encounter an interesting situation - this is a vertical line.
Vertical lines have an equation in the form \(x = a\). Since our equation does not depend on y, and only specifies where x is, there isn't a corresponding point where the line touches the y-axis. Thus, there is no y-intercept for the equation \(x = -3\).
In most linear equations (like \(y = mx + b\)), you can find the y-intercept by substituting 0 in place of x.
vertical line
Understanding vertical lines is crucial in graphing linear equations. A vertical line is a line that runs up and down the graph, parallel to the y-axis.
Its defining characteristic is that all points on the line have the same x-coordinate.
For example, the equation \(x = -3\) describes a vertical line where every point has an x-coordinate of -3.
Vertical lines do not have a slope (the concept of slope doesn't apply) and do not intersect the y-axis. To graph a vertical line:
  • Identify the x-value given in the equation (here, it is -3).
  • Plot a point at (-3, y) for any value of y.
  • Draw a straight line through these points.
The line will be parallel to the y-axis, passing through \((-3, y)\) for all y-values.

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Most popular questions from this chapter

Give an example of a function from everyday life. (Hint: Fill in the blanks:_____ depends on_____,\(\mathrm{SO}\)_____is a function of_____.)

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

Find an equation of the line passing through the given points. (a) Write the equation in standard form. (b) Write the equation in slope-intercept form if possible. $$ \left(\frac{3}{4}, \frac{8}{3}\right) \text { and }\left(\frac{2}{5}, \frac{2}{3}\right) $$

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. See Examples \(1-3\) $$ f\left(\frac{1}{3}\right) $$

Graph each equation with a graphing calculator. Use the standard viewing window. $$ 5 x+2 y=-10 $$
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