Problem 14 - Precalculus

Question

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=5 f(2 x+1)+3 $$

Step-by-Step Solution

Answer

The point \(\left(\frac{1}{2}, -12\right)\) is on the graph of the transformed function.

Step 1: Understand the Point Transformation

Given that the point \((2, -3)\) is on the graph of \(y=f(x)\), we need to find an equivalent point on the graph of the transformed function \(y=5f(2x+1)+3\). We start by identifying the variable transformations in the function.

Step 2: Horizontal Transformation

The transformation \(2x + 1\) suggests a horizontal shift. To find the corresponding x-coordinate on the graph of \(y = f(x)\), solve for \(x\) by substituting \(x = 2\) into the equation:\[2x + 1 = 2\]Subtract 1 from both sides:\[2x = 1\]Divide by 2:\[x = \frac{1}{2}\]

Step 3: Evaluate the Function at the Transformed x-value

With \(x = \frac{1}{2}\), find the corresponding function value. Because \((2, -3)\) is on the graph of \(y = f(x)\), we have:\[f(2) = -3\]We now consider any vertical transformation next.

Step 4: Vertical Scaling and Translation

Insert \(f(2x+1) = f(2)\) into the transformed function:\[y = 5f(2x+1) + 3 = 5(-3) + 3\]Simplify the expression:\[-15 + 3 = -12\]

Step 5: Combine Results

The transformed \(x\)-coordinate is \(\frac{1}{2}\) and the transformed \(y\)-coordinate is \(-12\). Therefore, the point \(\left(\frac{1}{2}, -12\right)\) is on the graph of the given function.

Key Concepts

Horizontal Transformation

Horizontal transformation involves shifting the graph of a function left or right on the coordinate plane. In this scenario, you have the expression \(2x + 1\) inside the function \(f\), which means your graph will undergo a horizontal transformation.

• The term \(2x\) indicates a horizontal compression factor. It means the graph is squeezed horizontally by a factor of 2. Here, each \(x\) is halved, leading to the x-value transformation, where \(x = 2\) becomes \(x = \frac{1}{2}\) after solving the equation \(2x + 1 = 2\).
• The plus 1 inside the expression \(2x + 1\) indicates a left shift by \(\frac{1}{2}\) unit. Instead of moving the graph to the right as you add inside, the graph actually shifts to the left because you solved for when the inside equals the original \(x\) value.

Hence, the original point \((2, -3)\) transforms to \(\left(\frac{1}{2}, -3\right)\) on the x-axis due to these changes.

Vertical Transformation

Vertical transformation changes the appearance of the graph by altering the vertical position and stretching or compressing it vertically. In this case, consider the function transformed to \(y = 5f(2x+1)+3\).

• The coefficient 5 multiplies the \(f(x)\), indicating a vertical stretch. This multiplication makes everything 5 times taller, multiplying the original function value by 5. Here, since \(f(2) = -3\), the scaling results in \(5 \times -3 = -15\).
• The addition of 3 after the function signifies a vertical shift upwards by 3 units. Thus, the final \(y\) coordinate is found by adding 3 to the scaled value: \(-15 + 3 = -12\).

By applying this vertical stretch and translation, the y-coordinate of the point on the transformed graph becomes \(-12\).

Point Transformation

Point transformation determines how the coordinates of a point on a function graph change under both vertical and horizontal transformations. With the original point \((2, -3)\), let's see how each transformation step affects the point's coordinates.

• Horizontal Change: The horizontal transformation affects only the x-value, turning \(2\) into \(\frac{1}{2}\). This comes from substituting the value into \(2x + 1\) and solving for \(x\).
• Vertical Change: Vertical transformation fits into the conclusions from the previous section, scaling and shifting \(y\) from \(-3\) to \(-12\).

The resulting transformed point becomes \(\left(\frac{1}{2}, -12\right)\), combining both the horizontal and vertical changes.

Graphing Functions

When graphing transformed functions, recognizing the combined effects of horizontal and vertical changes is essential. Each transformation shifts or stretches the original graph into its new position.

• Horizontal Moves: Adjusting for transformations like \(2x+1\), the entire graph shifts as per the solution derived, moving all x-coordinates correspondingly. You first solve for these x-coordinate transformations before adjusting the graph.
• Vertical Adjustments: Vertical transformations affect y-values globally. Scaling, as with multiplying by 5, adjusts the graph's height while translation by adding 3 shifts it up or down altogether.

Graphing these transformations overlays the point transformations onto the entire graph, altering its alignment and proportionalities based on given transformations. Recognize each component's effect to accurately reflect these changes in graph form.