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Chapter 7: Problem 103

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

Expert verified
The corresponding point is (6, -2).

Step by step solution

01

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).
02

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: \(3+3=6\).
03

Apply Vertical Scaling

The function also includes a vertical scaling factor of \(\frac{1}{2}\). Multiply the original y-coordinate (-4) by \(\frac{1}{2}\): \(-4 \times \frac{1}{2} = -2\).
04

Write the New Point

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle \(\theta\). $$\cos \theta=\frac{1}{3}$$

\(f(x)=\sin x, g(x)=\cos x, h(x)=2 x,\) and \(p(x)=\frac{x}{2} .\) Find the value of each of the following: $$ (h \circ f)\left(\frac{\pi}{6}\right) $$

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Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$\tan 10^{\circ} \cot 10^{\circ}$$

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