Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. \(f(x)=-|x-9|+16\)

Step-by-Step Solution

Answer

The x-intercepts are at \((25, 0)\) and \((-7, 0)\); the y-intercept is at \((0, 7)\).

Step 1: Find the y-intercept

To find the y-intercept, substitute \(x=0\) into the function and solve for \(f(x)\). \[ f(0) = -|0-9| + 16 = -|9| + 16 = -9 + 16 = 7 \] So, the y-intercept is at the point \((0, 7)\).

Step 2: Set the function equal to zero to find x-intercept(s)

To find the x-intercept(s), set the function equal to zero: \(-|x-9| + 16 = 0\). Solving this equation will give us the points where the graph intersects the x-axis. Rearranging, we get: \[ -|x-9| = -16 \] \[ |x-9| = 16 \]

Step 3: Solve for x-intercept(s)

The equation \(|x-9| = 16\) has two cases to consider since we are dealing with an absolute value:1. \(x-9 = 16\) - Solving gives: \(x = 25\)2. \(x-9 = -16\) - Solving gives: \(x = -7\) Thus, the x-intercepts of the graph are at the points \((25, 0)\) and \((-7, 0)\).

Key Concepts

x-intercept

An x-intercept is where a graph crosses the x-axis. At this point, the output value of the function is zero, meaning that the y-value is zero. To find the x-intercepts of a function, you must set the function equal to zero and solve for x.

In our exercise, the function is given by\[ f(x) = -|x-9| + 16 \] To find the x-intercepts, set \[ -|x-9| + 16 = 0 \] This results in the equation \[ |x-9| = 16 \].

The absolute value equation \[ |x-9| = 16 \] provides two scenarios:

• \(x-9 = 16\) which simplifies to \(x = 25\)
• \(x-9 = -16\) which simplifies to \(x = -7\)

These solutions indicate the x-intercepts at points \((25, 0)\) and \((-7, 0)\).

To sum up, finding x-intercepts requires solving the equation obtained by equaling the function to zero. This tells us where the graph intersects the x-axis.

y-intercept

A y-intercept is where the graph of a function crosses the y-axis. At this point, x is equal to zero. To find the y-intercept, you simply substitute x with zero in the function and solve for the output value.

In the given function \[ f(x) = -|x-9| + 16 \], substituting x with zero gives us: \[ f(0) = -|0-9| + 16 \]\[ = -|9| + 16 \] \[ = -9 + 16 \] \[ = 7 \]

So, the y-intercept is the point \((0, 7)\).

This point is crucial as it tells us where the function begins on the y-axis when graphing the function, providing an anchor for the graph's y-value.

absolute value function

An absolute value function is characterized by a V-shaped graph. It is defined mathematically by the expression inside the absolute value brackets, and consists of two linear pieces meeting at a vertex.

The function in our exercise is presented as\[ f(x) = -|x-9| + 16 \].

This means the graph will be flipped upside down (or reflected across the x-axis) and shifted vertically upwards by 16 units due to the '-1' coefficient and the '+16' constant, respectively. The absolute value \[ |x-9| \]ensures that the expression inside forms the base of the two linear segments, centered at \(x = 9\).

Understanding an absolute value function involves grasping its reflective symmetry, vertical shifts, and impact of coefficients.

step-by-step solution

A step-by-step solution assists in breaking down complex problems into digestible parts. Each step is carefully followed, helping students see how one part leads logically to the next and ensuring they understand every component of the process.

In our exercise, we first find the y-intercept by substituting x = 0 as:

• Calculate \( f(0) = -|0-9| + 16 \)
• Solve to get the y-intercept \((0, 7)\)

This helps students visualize what happens when x is zero.

Next, we solve for the x-intercepts by setting \(-|x-9| + 16\) to zero, unfolding it into:

• \(-|x-9| = -16\)
• Breaking the absolute value into two equations, \(x-9 = 16\) and \(x-9 = -16\), yields solutions \(x = 25\) and \(x = -7\)

Breaking the solution into steps emphasizes understanding each part, providing thorough insight into both intercepts and function behavior.