Question

Suppose that the \(x\) -intercepts of the graph of \(y=f(x)\) are -5 and 3 . (a) What are the \(x\) -intercepts of the graph of \(y=f(x+2) ?\) (b) What are the \(x\) -intercepts of the graph of \(y=f(x-2) ?\) (c) What are the \(x\) -intercepts of the graph of \(y=4 f(x) ?\) (d) What are the \(x\) -intercepts of the graph of \(y=f(-x) ?\)

Short answer

For (a), x-intercepts are -7, 1. For (b), they are -3, 5. For (c), they remain -5, 3. Finally, for (d), they change to 5, -3.

Step-by-step solution

Understand the Problem

The given problem involves finding the new x-intercepts of the function when it undergoes various transformations. The original x-intercepts are at \(x = -5\) and \(x = 3\).

Determine the x-intercepts for y = f(x+2)

The transformation \(x + 2\) translates the graph horizontally to the left by 2 units. Therefore, if the original x-intercepts are \(x = -5\) and \(x = 3\), the new x-intercepts will be \(x = -5 - 2 = -7\) and \(x = 3 - 2 = 1\).

Determine the x-intercepts for y = f(x-2)

The transformation \(x - 2\) translates the graph horizontally to the right by 2 units. Hence, the new x-intercepts will be \(x = -5 + 2 = -3\) and \(x = 3 + 2 = 5\).

Determine the x-intercepts for y = 4f(x)

The transformation \(4f(x)\) scales the graph vertically but does not affect the horizontal positions of the x-intercepts. Therefore, the x-intercepts remain \(x = -5\) and \(x = 3\).

Determine the x-intercepts for y = f(-x)

The transformation \(f(-x)\) reflects the graph across the y-axis. The x-intercepts will therefore be \(x = 5\) and \(x = -3\), as the original values are negated.

Key concepts

Graph Transformations

Graph transformations are changes made to the graph of a function that alter its shape or position. These transformations can be applied in various ways, such as moving the graph horizontally, scaling it vertically, or reflecting it across an axis. Understanding these transformations is crucial because they help us see how the function behaves under different conditions. Common transformations include horizontal translations, vertical scaling, and reflections.

Horizontal Translation

Horizontal translation shifts the entire graph of a function either to the left or right. This is done by adding or subtracting a constant value to the variable inside the function. For example, in the function \(y = f(x + 2)\), the graph shifts to the left by 2 units. This means that every point on the graph moves 2 units to the left. If the original x-intercepts were at -5 and 3, the new x-intercepts will be at -7 and 1.
Alternatively, in \(y = f(x - 2)\), the graph shifts to the right by 2 units. So, the new x-intercepts will be at -3 and 5. Horizontal translations are straightforward once you understand the direction of the shift based on the sign of the constant.

Vertical Scaling

Vertical scaling involves multiplying the entire function by a constant. This alters the amplitude of the graph without changing its x-intercepts. For instance, in the function \(y = 4f(x)\), every y-value is scaled by a factor of 4. This makes the graph steeper or 'stretched' vertically, but the points where the graph intersects the x-axis (the x-intercepts) remain the same. In our example, the x-intercepts will still be at -5 and 3 regardless of the scaling factor.

Reflection Across Y-Axis

Reflecting a graph across the y-axis changes the direction of the graph, creating a mirror image. This is done by replacing the variable x with -x in the function. For example, in \(y = f(-x)\), every x-value is negated; thus, if the original x-intercepts were at -5 and 3, they will change to 5 and -3 after reflecting. The graph looks like it has been flipped over the y-axis, changing the signs of the x-intercepts while maintaining their absolute values.