Math Problem Statement

Describe verbally the transformations that can be used to obtain the graph of g from the graph of f. ​g(x)equals=7 Superscript x plus 97x+9​; ​ f(x)equals=7 Superscript x7x Question content area bottom Part 1 Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) within your choice. ​(Type integers or simplified​ fractions.) A. The graph of g is the graph of f shifted enter your response here ​unit(s) to the right and stretched horizontally by a factor of enter your response here. B. The graph of g is the graph of f shifted enter your response here ​unit(s) to the left and stretched horizontally by a factor of enter your response here. C. The graph of g is the graph of f shifted enter your response here ​unit(s) up. D. The graph of g is the graph of f reflected in the​ y-axis. E. The graph of g is the graph of f shifted enter your response here ​unit(s) down and stretched horizontally by a factor of enter your response here. F. The graph of g is the graph of f shifted enter your response here ​unit(s) to the leftleft. G. The graph of g is the graph of f reflected in the​ x-axis. H. The graph of g is the graph of f shifted enter your response here ​unit(s) down. I. The graph of g is the graph of f shifted enter your response here ​unit(s) up and stretched horizontally by a factor of enter your response here. J. The graph of g is the graph of f shifted enter your response here ​unit(s) to the rightright.

Solution

Let's analyze the transformation from the graph of f(x)=7xf(x) = 7^x to the graph of g(x)=7x+9g(x) = 7^x + 9.

Step-by-step explanation:

  • The function f(x)=7xf(x) = 7^x is an exponential function where the base is 7.
  • The function g(x)=7x+9g(x) = 7^x + 9 adds a constant 99 to the original function f(x)f(x).

Transformation:

Adding a constant to the function causes a vertical shift of the graph. Specifically:

  • The function g(x)=7x+9g(x) = 7^x + 9 is the graph of f(x)=7xf(x) = 7^x shifted 9 units upward.

So, the correct transformation is a vertical shift of the graph of f(x)f(x) by 9 units up.

The correct choice is: C. The graph of gg is the graph of ff shifted 9 units up.

Would you like further details or have any questions? Here are some related questions:

  1. How do transformations affect the horizontal or vertical asymptotes of exponential functions?
  2. What happens if you add a negative constant to an exponential function?
  3. How does multiplying the exponent affect the graph of an exponential function?
  4. What is the general form of an exponential function transformation?
  5. How does the base of an exponential function affect the steepness of the graph?

Tip: Vertical shifts only affect the y-values of the graph, without changing the shape or the horizontal positions of the graph.

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Math Problem Analysis

Mathematical Concepts

Exponential Functions
Graph Transformations

Formulas

g(x) = f(x) + c (Vertical Shift Formula)
f(x) = 7^x (Exponential Function)

Theorems

Vertical Shift Theorem

Suitable Grade Level

Grades 9-11

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