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Chapter 1: Problem 21

Identify the \(x\) - and \(y\) -intercepts of the graph. Verify your results algebraically. (GRAPH CAN'T COPY) $$y=|x+2|$$

Short Answer

Expert verified
The \(x\)-intercept is -2 and the \(y\)-intercept is 2.

Step by step solution

01

Find the \(x\)-intercept

To find the \(x\)-intercept, set \(y=0\). The equation then becomes \(0 = |x+2|\). In order for this to be true, the expression inside the absolute value symbols must be zero, because the absolute value of a number is never negative. So \(x + 2 = 0\), and solving this we get \(x = -2\) as the \(x\)-intercept.
02

Find the \(y\)-intercept

To find the \(y\)-intercept, set \(x=0\). The equation then becomes \(y = |0+2|\), so \(y = |2|\) or \(y = 2\). Thus, the \(y\)-intercept is 2.
03

Verify the results

To verify these results algebraically for \(x = -2\), substitute \(x = -2\) into the given \(y = |x+2|\) equation: \(y = |-2 + 2| = |0| = 0\). As for \(y = 2\), substitute \(x = 0\) into the equation: \(2 = |0 + 2| = |2| = 2\). In both substitutions, the results are consistent, so the \(x\) - and \(y\) -intercepts are correct.

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Key Concepts

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

Understanding the X-Intercept
The x-intercept is the point where a graph crosses the x-axis. For any point on the x-axis, the y-coordinate is always zero. Therefore, to find the x-intercept of an equation, we set y equal to zero and solve for x.

In our problem with the equation \(y = |x+2|\), we want to find the x-coordinate when y is equal to zero. So we adjust the equation to \(0 = |x+2|\). Since the absolute value of any number is non-negative, the inside of the absolute value must be zero for this equation to be true.
  • Solving \(x + 2 = 0\), we find \(x = -2\).
  • Therefore, the x-intercept is the point \((-2, 0)\).
This means that when we plot this function, it will cross the x-axis at x = -2.
Pinpointing the Y-Intercept
The y-intercept is the point where a graph crosses the y-axis. On the y-axis, all points have an x-coordinate of zero. To find the y-intercept, we set x to zero and solve for y in the equation.

For the equation \(y = |x+2|\), we substitute \(x = 0\) to find the y-value. This gives us \(y = |0+2|\), which simplifies to \(y = |2| = 2\).
  • Hence, the y-intercept is the point \((0, 2)\).
This tells us that as the graph crosses the y-axis, it does so at y = 2.
Exploring the Absolute Value Function
The absolute value function is a crucial concept in precalculus. The absolute value of a number is its distance from zero on the number line, regardless of direction, meaning it is always non-negative.

An equation involving absolute values, like \(y = |x+2|\), means that for every x, the output will be the non-negative version of \(x + 2\). If the expression inside the absolute value is negative, the function will output its positive equivalent. This nature creates a distinctive V-shape graph.
  • If \(x + 2\) is positive or zero, the graph moves linearly upward.
  • If \(x + 2\) is negative, the graph reflects upward from the x-axis.
This property significantly shapes the graph, providing solutions like our x-intercept and y-intercept as important waypoints along this mathematical path.

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Most popular questions from this chapter

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