Question

If the point (3,-4) is on the graph of \(y=f(x),\) what corresponding point must be on the graph of \(\frac{1}{2} f(x-3) ?\)

Short answer

The corresponding point is (6, -2).

Step-by-step solution

Identify the Original Point

The given point is (3, -4). This means when x = 3, the function value is y = -4 for the graph of y = f(x).

Apply Horizontal Shift

The function \(\frac{1}{2} f(x-3)\) includes a horizontal shift. The term \((x-3)\) indicates a right shift by 3 units. To find the new x-coordinate, add 3 to the original x-coordinate: \(3+3=6\).

Apply Vertical Scaling

The function also includes a vertical scaling factor of \(\frac{1}{2}\). Multiply the original y-coordinate (-4) by \(\frac{1}{2}\): \(-4 \times \frac{1}{2} = -2\).

Write the New Point

Combining the results of the previous steps, the new point is (6, -2).

Key concepts

Horizontal Shift

A horizontal shift is a transformation that moves a graph left or right. It's determined by the input of the function, specifically when x is adjusted by a constant value.

For example, in the function \(f(x-3)\), the \(-3\) inside the parenthesis implies that every x-value on the graph of \(f(x)\) will be shifted to the right by 3 units.

• Positive values inside the function \(f(x+c)\) shift the graph to the left.
• Negative values inside the function \(f(x-c)\) shift the graph to the right.

So, if the original point was (3, -4) and the function becomes \(f(x-3)\), the new x-coordinate would be \(3 + 3 = 6\).

Remember, horizontal shifts don't change the y values of the points, only the x values are affected.

Vertical Scaling

Vertical scaling is a transformation that stretches or compresses the graph of a function in the y-direction. It involves multiplying the function by a constant factor.

Given a function \(\frac{1}{2} f(x)\), the graph of \(f(x)\) is scaled vertically by a factor of \(\frac{1}{2}\). This means every y-value is multiplied by \(\frac{1}{2}\).

• Factors greater than 1, such as 2\(f(x)\), stretch the graph upwards.
• Factors between 0 and 1, such as \(\frac{1}{2} f(x)\), compress the graph towards the x-axis.

For our specific point (3, -4), applying a vertical scaling of \(\frac{1}{2}\) means:

Original y-coordinate = -4
New y-coordinate = -4 \times \frac{1}{2} = -2

This transformation changes the y-values but leaves the x-values unchanged.

Graphing Functions

Graphing functions accurately is crucial for visualizing and understanding transformations. Here are the basic steps to follow:

• Identify the parent function: Start with the basic form of the function, such as \(f(x)\).
• Apply transformations: Implement shifts, scalings, reflections, or stretches as given.
• Plot key points: Mark essential points on the graph, especially any transformed points.
• Draw the graph: Connect the points smoothly, ensuring the shape reflects all transformations.

In our given example, starting from the point (3, -4) on the graph of \(y=f(x)\), we perform the transformations:

1. Shift right by 3 units, changing the x-value from 3 to 6.
2. Scale vertically by \(\frac{1}{2}\), changing the y-value from -4 to -2.

The new point after these transformations is (6, -2). Plotting and connecting such points helps create an accurate representation of the function.

Algebra

Algebra plays a vital role in understanding function transformations. It provides the foundational operations needed for manipulating and transforming functions.

• Understanding expressions: Grasp how changes within functions, such as \(f(x-3)\), affect the graph.
• Basic calculations: Accurately perform arithmetic operations, such as multiplication for vertical scaling.
• Analyzing functions: Break down complex functions into simpler parts to understand transformations.

In our problem, algebra helps us:

1. Determine the horizontal shift by solving \(x-3 = 0\) to get \x = 3\.
2. Calculate the new y-value by multiplying the original y-value by the vertical scaling factor \(\frac{1}{2}\).

These steps require a solid understanding of algebraic manipulations which ensure accurate transformations and graph plotting.