Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the fact that the graph of \(y=f(x)\) has \(x\) -intercepts at \(x=2\) and \(x=-3\) to find the \(x\) -intercepts of the given graph. If not possible, state the reason.$$y=f(x)+2$$.

Step-by-Step Solution

Answer

The x-intercepts of the function \(y=f(x)+2\) are still at \(x=2\) and \(x=-3\), but instead of the function value being zero at these points, it is now -2 because of the upward shift of the function.

Step 1: Understanding the Relationship Between the Functions

This problem involves a parent function \(f(x)\) and a function that is a variation of it, \(y=f(x)+2\). The latter function is the original function but shifted upward by two units.

Step 2: Identifying Original x-Intercepts

The original function \(f(x)\) has x-intercepts at \(x=2\) and \(x=-3\). This means when \(x=2\) or \(x=-3\), \(y=f(x)\) equals to zero.

Step 3: Applying the Vertical Shift

A vertical shift does not change the x-intercepts of a function. This is because a vertical shift involves adding or subtracting a constant to the output of the function (y-value), not the input (x-value). Thus, the x-intercepts, where \(y=0\), would remain the same even after the shift.

Step 4: Determining the x-Intercepts of the Shifted Function

For the shift function \(y=f(x)+2\), the x-intercepts can be found by setting \(y\) to zero and solving for \(x\). However, since there is a vertical shift of +2, the x-intercepts of the parent function \(f(x)\) when \(y=0\) will now become the x-intercepts of \(y=f(x)+2\) when \(y=-2\). Therefore, the x-intercepts of the function \(y=f(x)+2\) will also be at \(x=2\) and \(x=-3\), but the function value at these points will be -2 instead of 0.

Key Concepts

Function Transformations

Function transformations refer to changes in the function's graph, such as shifts, reflections, stretches, or compressions. These transformations can alter how a function's graph appears without changing the fundamental structure or nature of the function itself.

In the case of a vertical shift, you're simply moving the entire graph of the function up or down a certain number of units on the y-axis. A vertical shift is described mathematically as adding or subtracting a constant to the function. For example, in the transformation from \(y=f(x)\) to \(y=f(x)+2\), the whole graph moves up by two units.

Because a vertical shift affects the outputs (or y-values) and not the inputs (or x-values), the vertical shift doesn't alter where the graph crosses the x-axis. Understanding these transformations is vital for predicting changes to a graph when alterations are applied to the original function.

X-Intercepts

X-intercepts are points where a graph crosses the x-axis. At these points, the y-value of the function is zero. Finding these intercepts is crucial because they provide valuable insight into the function's roots or solutions.

When a function is vertically shifted, as in the scenario with \(y=f(x)+2\), the x-intercepts of the new function may seem confusing. However, remember that a simple vertical shift will not change the x-coordinates of where the graph intersects the x-axis. Instead, it changes the y-values at these points.

In our example, the function \(f(x)\) originally has x-intercepts at \(x=2\) and \(x=-3\). After the vertical shift of +2, the graph doesn't cross the x-axis at these points, but it "crosses" the line \(y=-2\) instead. Thus, while the x-intercepts are found in the same locations, the value of the function at these points is \(y=-2\).

Graphing Functions

Graphing functions is a powerful way to visually represent the behavior of equations. It helps in understanding their properties, such as intercepts, shifts, and general shapes.

When graphing, it’s important to plot known points and interpret transformations like shifts or reflections appropriately. For example, in the original graph of \(y=f(x)\) with x-intercepts at \(x=2\) and \(x=-3\), the task is to redraw this graph when vertical shifts occur.

• Start by identifying key features such as intercepts, maximum and minimum points.
• Apply transformations; in this case, shift up by 2 units.
• Coordinate adjustments based on transformations help maintain the correct shape and relationships of the function.

By understanding function transformations from a graphical perspective, students can confidently sketch functions and predict changes resulting from different modifications.