Problem 12 - College Algebra

Question

For the following exercises, find the \(x\) - and \(y\) -intercepts of the graphs of each function. $$ f(x)=-5|x+2|+15 $$

Step-by-Step Solution

Answer

x-intercepts at \((1, 0)\) and \((-5, 0)\); y-intercept at \((0, 5)\).

Step 1: Understanding the Problem

The goal is to find the points where the given function intersects the x-axis and y-axis. These are called the x-intercept and y-intercept, respectively.

Step 2: Finding the x-intercepts

To find the x-intercepts, we set the function equal to zero and solve for x:\[-5|x+2|+15 = 0\]Subtract 15 from both sides:\[-5|x+2| = -15\]Divide both sides by -5:\[|x+2| = 3\]This gives two equations:\[ x+2 = 3 \] and \[ x+2 = -3 \].Solving these gives:\[ x = 1 \] and \[ x = -5 \].Thus, the x-intercepts are \(x = 1\) and \(x = -5\).

Step 3: Finding the y-intercept

To find the y-intercept, we set x to zero and solve for f(x):\[f(0) = -5|0+2|+15\]Simplify:\[f(0) = -5(2) + 15 = -10 + 15 = 5\]Thus, the y-intercept is \( (0, 5) \).

Key Concepts

x-intercept

When dealing with graphing functions, the **x-intercept** is a key concept to understand. An x-intercept is where the graph of the function crosses the x-axis. This is the point where the output of the function, or y-value, is zero. In simpler terms, it's where the graph "touches" or "hits" the x-axis.

To find the x-intercept of a function, we need to set the entire function equal to zero and solve for x. For the function given in our exercise, we start by solving:

• Set the function to zero: \[ -5|x+2|+15 = 0 \]
• Subtract 15 from both sides: \[ -5|x+2| = -15 \]
• Divide by -5: \[ |x+2| = 3 \]

Since we have an absolute value, this equation splits into two possibilities:

• \[ x+2 = 3 \]
• \[ x+2 = -3 \]

Solving for x gives us:

• \[ x = 1 \]
• \[ x = -5 \]

Therefore, the x-intercepts of the function are at the points (1,0) and (-5,0).

y-intercept

Another essential concept in graphing functions is the **y-intercept**. The y-intercept is where the graph crosses the y-axis, meaning this is the point where the x-value is zero. When we're looking for the y-intercept, we're essentially asking, "What is the value of the function when x is 0?"

For the exercise's function, we find the y-intercept by setting x to 0 and evaluating the function:

• Set x to 0 in the function: \[ f(0) = -5|0+2|+15 \]
• Simplify inside the absolute value: \[ f(0) = -5(2) + 15 \]
• Calculate to find: \[ f(0) = -10 + 15 = 5 \]

Thus, the y-intercept is at the point (0, 5). This point helps to anchor our graph along the y-axis and gives a visual starting point when sketching the function.

absolute value function

The **absolute value function** is a crucial piece of understanding how the x-intercepts are calculated in our problem. The absolute value of a number is its distance from zero on the number line, regardless of direction, hence always positive. This unique property impacts how we solve equations involving absolute values, like in our given function:

The exercise starts with the expression \[ |x+2| \], a part of the absolute value function. Solving an equation with absolute values, as seen in \[ -5|x+2|+15 = 0 \], involves two cases because \[ |a| = b \] implies \[ a = b \] or \[ a = -b \]. This is why we split into two scenarios: \[ x+2 = 3 \] and \[ x+2 = -3 \].

Absolute value functions often present as a V-shape when graphed. The vertex of this V can shift based on the values inside the function and the coefficients. In our case, the function \[ f(x) = -5|x+2| + 15 \] takes an inverted V due to the negative coefficient \[ -5 \], causing a reflection over the x-axis. This property is fundamental when sketching or interpreting the graphs of absolute value functions in mathematics.