Problem 27 - Precalculus with Limits

Question

In Exercises 23-32, find the \( x \)- and \( y \)-intercepts of the graph of the equation. \( y = |3x-7| \)

Step-by-Step Solution

Answer

The y-intercept of the equation is 7, and the x-intercept is \(7/3\).

Step 1: Find the y-intercept

To find the y-intercept, we set \(x = 0\) in the equation and solve for \(y\). So, substituting \(x = 0\) in the equation \(y = |3x - 7|\), we get \(y = |-7|\), which simplifies to \(y = 7\). So, the y-intercept is 7.

Step 2: Find the x-intercept

To find the x-intercept, we set \(y = 0\) in the equation and solve for \(x\). This means we set \(|3x - 7| = 0\). An absolute value equals zero only when the expression inside is 0. This gives us \(3x - 7 = 0\). Solving for \(x\) gives \(x = 7/3\). So, the x-intercept is \(7/3\).

Key Concepts

Absolute Value Function

Absolute value functions have a unique characteristic. They make any number within the absolute value sign positive. If the function is written as \( y = |3x-7| \), it affects the graph in special ways. It means that whatever the value of \( 3x-7 \) becomes, the output \( y \) will always end up being non-negative. Absolute value functions typically form a "V" shape on a graph. This "V" is symmetrical around a certain point and changes direction at the point where the expression inside the absolute value equals zero.

This point is crucial for understanding the behavior of the graph. To find where it might change direction, we can solve \( 3x - 7 = 0 \). The solution, which is \( x = \frac{7}{3} \), helps us determine this vertex point.

In our current problem, the vertex is at \( (\frac{7}{3}, 0) \). It's the point where the graph switches from decreasing to increasing. Understanding this property helps in graphing absolute value functions efficiently.

X-Intercept

X-intercepts help us identify where the graph crosses the x-axis. For the function \( y = |3x-7| \), finding the x-intercept involves setting the function equal to zero. This is because at any x-intercept, \( y \) must be zero.

To find the x-intercept, we set the equation \( |3x - 7| = 0 \). Absolute values become zero when their insides are zero, so solve \( 3x - 7 = 0 \). This equation simplifies to \( x = \frac{7}{3} \).

Thus, \( (\frac{7}{3}, 0) \) is the x-intercept. It's where the graph touches the x-axis and is a key point when sketching the graph of an absolute value function.

Y-Intercept

The y-intercept reveals where the function's graph crosses the y-axis. For the absolute value function \( y = |3x-7| \), you find the y-intercept by setting \( x = 0 \).

By plugging \( x = 0 \) into the equation, we evaluate it as \( y = |3 \cdot 0 - 7| \). This simplifies to \( y = | -7 | \), which equals 7. Therefore, the y-intercept of this function is \( (0, 7) \).

The y-intercept is vital for creating a precise graph of any equation, giving you the starting point of the graph's journey across the axes.