Problem 105 If the graph of \(y=f(x)\) under... [FREE SOLUTION]

Chapter 3: Problem 105

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.

Short Answer

Expert verified

Same x-intercepts, different y-intercepts.

Step by step solution

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.

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.

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

91影视

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.

Short Answer

Step by step solution

Understanding Vertical Transformations

Analyzing Effects on x-Intercepts

Analyzing Effects on y-Intercepts

Key Concepts

Vertical Stretch

Vertical Shrink

x-intercepts

y-intercepts

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