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Chapter 1: Problem 28

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$y=-|x+10|$$

Short Answer

Expert verified
The x-intercept for the graph of the function \(y = - |x+10|\) is (-10, 0) and the y-intercept is (0, -10).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y = 0\) as the x-intercept is the point where the function crosses the x-axis. For this equation, the x-intercept is found when \( y = - |x + 10| = 0\). Therefore, \(|x + 10| = 0\). The absolute value of an expression equals zero only when the expression itself is zero, so \(x + 10 = 0\) which gives us \(x = -10\).
02

Find the y-intercept

The y-intercept is the point at which the line crosses the y-axis, this occurs when \(x = 0\). Substituting zero for x in the equation \$y = - |x + 10| gives us \(y = - |0 + 10| = -10\). Therefore, the y-intercept is located at (0,-10).
03

Summarize the intercepts

Summarize what you've found above. For the given function $y = - |x + 10|$, we have the x-intercept at (-10, 0) and the y-intercept at (0, -10).

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Key Concepts

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

Absolute Value Equations
An absolute value equation is an equation where the unknown variable is inside an absolute value operation, such as |x| = a. The absolute value of a number is its distance from zero on the number line, regardless of direction. As a result, these equations can have two possible solutions, since both x = a and x = -a would have the same absolute value. However, in our exercise, the solution is uniquely zero because only |x + 10| = 0 leads us directly to x = -10. This single solution arise because the absolute value expression equals to zero, which signifies that the distance to zero is also zero; hence, the term inside the absolute value must itself be zero.
Graphing Linear Equations
Graphing linear equations involves plotting the solutions of the equation on a coordinate grid to form a straight line. The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. However, the given equation y = -|x + 10| is not in the slope-intercept form due to the absolute value. This graph would consist of two rays which start from the y-intercept and go in opposite directions away from the x-axis - a V-shape. One important aspect in graphing is finding points clearly defined by the equation, such as intercepts, which are the points where the graph crosses the axes - these act as crucial guidelines for drawing the graph.
Intercepts of a Function
Intercepts of a function are specific points where the graph of the function crosses the axes. The x-intercept occurs where the graph crosses the x-axis, hence the y-value is zero. To find it, set y = 0 and solve for x. Conversely, the y-intercept is where the graph crosses the y-axis, where x = 0. It reflects where the function has its output when the input is zero. For the example equation y = - |x + 10|, we found the x-intercept to be at (-10, 0) and the y-intercept to be at (0, -10). These intercepts provide valuable starting points for graphing the function and understanding its behavior.
Solving Equations
Solving equations is a foundational skill in algebra. It involves finding values for the variables that make the equation true. The strategy for solving an equation can vary depending on the type of equation and its complexity. For linear equations, this typically means isolating the variable on one side of the equation. In the context of our problem, we set each intercept to zero in turn to isolate and solve for the other variable. Solving equations often requires understanding and applying properties of equality and operations, which, when used correctly, can simplify an equation to a point where the solution becomes evident, as was the case with our absolute value equation.

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Most popular questions from this chapter

Use the fact that 13 inches is approximately the same length as 33 centimeters to find a mathematical model that relates centimeters \(y\) to inches \(x .\) Then use the model to find the numbers of centimeters in 10 inches and 20 inches.

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Use the given values of \(k\) and \(n\) to complete the table for the inverse variation model \(y=k x^{n} .\) Plot the points in a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|}\hline x & 2 & 4 & 6 & 8 & 10 \\\\\hline y=k / x^{n} & & & & & \\\\\hline\end{array}$$ $$k=20, n=2$$

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