Problem 105 - College Algebra with Modeling and Visualization

Question

If the graph of \(y=f(x)\) undergoes a vertical stretch or shrink to become the graph of \(y=g(x),\) do these two graphs have the same \(x\) -intercepts? \(y\) -intercepts? Explain your answers.

Step-by-Step Solution

Answer

Same x-intercepts, different y-intercepts.

Step 1: Understanding Vertical Transformations

When a function undergoes a vertical stretch or shrink, the graph of the function is multiplied by a constant factor. For example, if we have a function \( y = f(x) \) and it gets transformed to \( y = a f(x) \), where \( a \) is a constant, this represents a vertical stretch if \( |a| > 1 \) or a shrink if \( 0 < |a| < 1 \). This transformation affects the distance of points from the \( x \)-axis but does not change their \( x \)-coordinates.

Step 2: Analyzing Effects on x-Intercepts

The \( x \)-intercepts of the graph \( y = f(x) \) occur where \( f(x) = 0 \). When we apply a vertical stretch or shrink to get \( y = g(x) = a f(x) \), the condition for \( x \)-intercepts becomes \( a f(x) = 0 \). Because \( a eq 0 \), this is equivalent to \( f(x) = 0 \), meaning the \( x \)-intercepts remain unchanged.

Step 3: Analyzing Effects on y-Intercepts

The \( y \)-intercept of the graph is where the graph crosses the \( y \)-axis, which occurs at \( x = 0 \). For the original function \( y = f(x) \), the \( y \)-intercept is \( f(0) \). After transformation, the intercept becomes \( g(0) = a f(0) \). Thus, the \( y \)-intercept will change by the factor \( a \) due to the vertical stretch or shrink.

Key Concepts

Vertical Stretch

A vertical stretch occurs when the graph of a function is elongated in the vertical direction by a factor greater than one. Imagine pulling the graph upwards or downwards. Mathematically, if we have a function expressed as \( y = f(x) \), and we apply a transformation \( y = a f(x) \) with \( |a| > 1 \), it's considered a vertical stretch. This means that every point on the graph will appear "stretched" farther from the x-axis than before.

Some key points to remember:

• The graph gets taller, but the horizontal positions of the points remain unchanged.
• This transformation impacts how the graph looks, making the peaks taller and the valleys deeper.

It's important to note that this does not affect the x-coordinates of the intercepts, as the stretching only occurs vertically.

Vertical Shrink

A vertical shrink occurs when the entire graph of a function is compressed towards the x-axis. This happens when the graph is scaled by a factor between 0 and 1. For example, if \( y = f(x) \) is transformed into \( y = a f(x) \) with \( 0 < |a| < 1 \), this indicates a vertical shrink.

Here are some important aspects:

• The distances from the x-axis are reduced, making the graph appear flatter.
• All y-values are closer to the x-axis compared to the original function.

Despite this compression in the vertical direction, the x-intercepts don't change because the transformation doesn't alter the x-values of the graph.

x-intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. In simpler terms, it's where the value of y becomes zero. For a function \( y = f(x) \), these are found by solving \( f(x) = 0 \).

During a vertical transformation, such as a stretch or shrink, the x-intercepts remain the same. This is because the condition for x-intercepts being \( f(x) = 0 \) does not change with the transformation to \( y = a f(x) \).

Why they remain unchanged:

• The factor \( a \) multiplying \( f(x) \) does not affect where the function equals zero.
• Both the original and the transformed functions will have their zeros at the same x-values.

In summary, the x-intercepts depend solely on the solutions to \( f(x) = 0 \), not the vertical scaling involved.

y-intercepts

Y-intercepts are the points where the graph crosses the y-axis. This happens when \( x = 0 \). At this point, for a function \( y = f(x) \), the y-intercept is simply \( f(0) \).

When a vertical transformation like a vertical stretch or shrink is applied, it affects the y-intercept. For the transformed graph \( y = g(x) = a f(x) \), the y-intercept will be at \( a f(0) \).

Key impacts:

• A vertical stretch will multiply the y-intercept by a factor greater than 1, making it larger in magnitude (further from zero).
• A vertical shrink will multiply the y-intercept by a factor less than 1, making it smaller in magnitude (closer to zero).

Thus, unlike x-intercepts, y-intercepts are directly influenced by the vertical scaling factor, changing location relative to their original position.