Problem 28 - Precalculus with Limits

Question

In Exercises 23-32, find the \( x \)- and \( y \)-intercepts of the graph of the equation. \( y = -|x+10| \)

Step-by-Step Solution

Answer

The x-intercept is \(x = -10\) and the y-intercept is \(y = -10\).

Step 1: Finding the x-intercept

We start by finding the x-intercept. The x-intercept is the value of \(x\) when \(y = 0\). Therefore, we'll first set up the equation with \(y\) equal to 0 and then solve for \(x\). The equation is \(0 = -|x + 10|\).

Step 2: Solving for x

The equation can be rewritten without the absolute value, it means the expression \(x + 10\) could be positive or negative, so we need to consider both cases: \(0 = -(x + 10)\) and \(0 = -(10 - x)\). Solving these equations will give us the x-intercept.

Step 3: Finding the y-intercept

Now, we move on to find the y-intercept. The y-intercept is the value of \(y\) when \( x = 0 \). Substituting \(x = 0\) into the original function, we obtain \( y = -|0+10| \).

Step 4: Solving for y

Finally, we solve the resulting equation from step 3, \( y = -|0+10| \), to find the y-intercept.

Key Concepts

x-intercept

The concept of an x-intercept is essential in understanding where a graph crosses the x-axis on a coordinate plane. For a point to lie on the x-axis, its y-value must be zero. This is why we set the equation equal to zero and solve for the x variable when looking for an x-intercept.

In our equation, \( y = -|x + 10| \), we find the x-intercept by setting \( y = 0 \), which gives us:

• \( 0 = -|x + 10| \)

This absolute value expression \(|x + 10|\) implies two cases we need to consider:

• \( x + 10 = 0 \) → solving gives \( x = -10 \)
• \( -(x + 10) = 0 \) → solving gives \( -x - 10 = 0 \), which can be rearranged to also yield \( x = -10 \)

Therefore, the x-intercept is at the point \( (-10, 0) \).

Remember, the x-intercept tells you where the graph hits the x-axis. Each equation may have one, multiple, or no x-intercepts depending on its nature.

y-intercept

Finding the y-intercept involves understanding where a graph touches the y-axis. This is the point where \( x = 0 \). Hence, for any equation, you substitute \( x = 0 \) to find the corresponding y-value. This y-value is the y-intercept.

Let's consider our absolute value function \( y = -|x + 10| \). To find the y-intercept, set \( x = 0 \) in the equation:

• \( y = -|0 + 10| \)

Simplifying the expression:

• \( y = -|10| = -10 \)

Therefore, the y-intercept occurs at the point \( (0, -10) \).

The y-intercept provides critical information about the graph's behavior, particularly where the graph crosses the y-axis. It's a straightforward yet powerful concept that's prevalent across linear, quadratic, and more complex equations like absolute value functions.

absolute value function

An absolute value function is an interesting mathematical expression. It revolves around \(|x|\), which denotes the distance of \(x\) from zero on a number line, always returning a non-negative result. This property of making any input non-negative is what gives the absolute value function its unique V-shape when graphed.

In the function \( y = -|x + 10| \), the minus sign in front of the absolute value inverts the V-shape to form an upside-down V. The expression inside the absolute value, \(x + 10\), shifts the graph horizontally. In this case, it moves 10 units to the left due to the \(+10\).

Understanding absolute value functions requires you to handle both the cases of the expression inside the absolute value being positive and negative. This is why when solving, we consider both scenarios, despite their implications being symmetric about the vertex point.

An absolute value function is visually recognizable and structurally unique in mathematics. Its reflection property and symmetry make it important in multiple mathematical contexts, including modeling situations where a value's magnitude matters more than its sign.