Question

If (1,3) is a point on the graph of \(y=f(x),\) which of the following points must be on the graph of \(y=2 f(x) ?\) (a) \(\left(1, \frac{3}{2}\right)\) (b) (2,3) (c) (1,6) (d) \(\left(\frac{1}{2}, 3\right)\)

Short answer

(c) (1, 6)

Step-by-step solution

- Understand the function transformation

The problem involves a vertical scaling of the function. The point (1,3) on the graph of the function y=f(x) is given. We need to find out how this transforms when the function is scaled by a factor of 2, making it y=2f(x).

- Apply the transformation

For a transformation of the form y=2f(x), every y-coordinate in y=f(x) is multiplied by 2. Hence, for the point (1,3) on the graph of y=f(x), the corresponding point on y=2f(x) would be (1, 2*3).

- Calculate the new point

Calculate 2*3, which equals 6. Therefore, the transformed point is (1,6).

- Match the result with the given choices

Check the given options: (a) (1, 3/2), (b) (2, 3), (c) (1, 6), (d) (1/2, 3). The point (1,6) matches option (c).

Key concepts

Vertical Scaling

Vertical scaling is an important concept in function transformations. When you scale a function vertically, you multiply its output values (y-values) by a constant factor. This doesn't affect the x-values. Instead, it stretches or compresses the graph up and down.
For example, if you have the function \( y = f(x) \) and you apply a vertical scaling by a factor of 2, the modified function is \( y = 2f(x) \).
This means every output value from \( f(x) \) is now twice as large. So if there is a point (1, 3) on the graph \( y = f(x) \), it transforms to (1, 6) on the graph \( y = 2f(x) \).
This is because the original y-value 3 is multiplied by 2, giving you 6. This understanding helps solve problems involving vertical scaling, like finding the corresponding points after a transformation.

Graph Transformations

Graph transformations involve changing the position or shape of a graph. These changes can occur in various ways, including translations, reflections, and scaling.
- **Translations** move the graph horizontally or vertically without changing its shape. For example, adding a constant to a function, like \( y = f(x) + c \), shifts the graph up by \( c \) units.
- **Reflections** flip the graph across a line, like the x-axis or y-axis. An example is \( y = -f(x) \), which reflects the graph across the x-axis.
- **Scaling** changes the size of the graph. Vertical scaling, which we've seen, multiplies the y-values. Horizontal scaling multiplies the x-values.
Knowing these transformations helps you understand how a graph of a function changes in response to different operations. In our exercise, vertical scaling was used to find the new point on the graph \( y = 2f(x) \) given the point on \( y = f(x) \).

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. To evaluate a function, substitute the input value into the function and calculate the result.
For instance, if you have \( f(x) = x^2 \) and want to evaluate it at \( x = 3 \), you substitute 3 for x: \( f(3) = 3^2 \). This yields 9.
In our exercise, the evaluation process was slightly different because we were dealing with a transformed function. Given the original point on the function \( y = f(x) \), we needed to evaluate the new function \( y = 2f(x) \).
For the point (1, 3) on y = f(x), the corresponding point on y = 2f(x) is evaluated by multiplying the original y-value by 2:
- *Original*: (1, 3)
- *Transformed*: 2 * 3 = 6
The new point is (1, 6).