Problem 106 - 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 unless \(f(0) = 0\).

Step 1: Understanding the Problem

We need to determine if the graphs of \(y=f(x)\) and \(y=g(x)\), where \(g(x)\) is obtained by vertically stretching or shrinking \(f(x)\), have the same \(x\)-intercepts and \(y\)-intercepts.

Step 2: Vertical Stretch or Shrink Definition

When we vertically stretch or shrink a graph, we transform \(f(x)\) into \(g(x) = c \cdot f(x)\), where \(c\) is a constant. If \(c > 1\), the graph is stretched. If \(0 < c < 1\), it's shrunk.

Step 3: Analyzing x-intercepts

The \(x\)-intercepts occur where \(y = 0\). For \(y = f(x)\), these occur at points \(x = a\) such that \(f(a) = 0\). Since multiplying by \(c\) (\(y = g(x) = c \cdot f(x)\)) doesn’t affect zeros, both the functions have the same \(x\)-intercepts.

Step 4: Analyzing y-intercepts

The \(y\)-intercept of a function is found by setting \(x=0\). For \(y=f(x)\), the \(y\)-intercept is \(f(0)\). For \(y=g(x) = c\cdot f(x)\), it becomes \(c \cdot f(0)\), which means they may differ unless \(f(0) = 0\).

Step 5: Conclusion

The graphs \(y = f(x)\) and \(y = g(x)\) will have the same \(x\)-intercepts because these points are where the function is zero, unaffected by vertical scaling. However, their \(y\)-intercepts will generally differ after a vertical stretch or shrink unless the original \(y\)-intercept is zero.

Key Concepts

Function Transformation

In the world of functions and graphs, transformations can seem magical. The term "function transformation" refers to any operation that alters the appearance or position of a graph without changing its basic shape.

When dealing with vertical stretches and shrinks, transformation becomes quite intriguing. These transformations occur when a constant, typically denoted by \(c\), multiplies the function. For example, if \(y = f(x)\), a vertical stretch or shrink changes it to \(y = g(x) = c \cdot f(x)\).

• If \(c > 1\), the function experiences a "stretch," making it appear taller.
• If \(0 < c < 1\), the function undergoes a "shrink," appearing shorter.

Such transformations adjust the height of the graph but not the horizontal position of any specific points, which leads into how vertical changes specifically impact intercepts.

x-intercepts

Understanding the concept of the \(x\)-intercepts is critical to analyzing how transformations affect a graph. The \(x\)-intercepts of a function are those specific points where the graph crosses the x-axis, meaning these are the points where the value of \(y\) equals zero.

For the original function \(y = f(x)\), these intercepts are found wherever \(f(a) = 0\). When the function is transformed to \(y = g(x) = c \cdot f(x)\), the \(x\)-intercepts remain unchanged because multiplying \(f(x)\) by any constant \(c\) doesn't alter the points where the function equals zero.

• This means that vertical stretching or shrinking has no effect on \(x\)-intercepts.
• Thus, \(y = f(x)\) and \(y = g(x)\) will share the same \(x\)-intercepts, maintaining their positions on the x-axis.

The consistency of x-intercepts during vertical transformations is one of the reasons these concepts can be easier to intuitively grasp.

y-intercepts

In contrast to \(x\)-intercepts, the \(y\)-intercepts can be affected by vertical transformations. The \(y\)-intercept of a graph is the point where it crosses the y-axis, determined by evaluating the function at \(x = 0\).

For the function \(y = f(x)\), the \(y\)-intercept is \(f(0)\). With the transformed function \(y = g(x) = c \cdot f(x)\), this point changes to \(c \cdot f(0)\).

• If \(f(0)\) is not zero, the y-intercept will change based on the value of \(c\).
• The y-intercept will remain the same only if the original value \(f(0)\) was zero.

This highlights an essential difference between \(x\)-intercepts and \(y\)-intercepts during transformation: \(y\)-intercepts are more sensitive to vertical stretching or shrinking, making them a pivotal point of interest in function transformation discussions.