Math Problem Statement

The table shows three values of x and their corresponding values of g(x), where g(x) = f(x) / (x + 3), and f is a linear function. What is the y-intercept of the graph of y = f(x) in the xy-plane?

Solution

We are given a table showing values of xx and their corresponding values of g(x)g(x), where g(x)=f(x)x+3g(x) = \frac{f(x)}{x+3}, and we know that f(x)f(x) is a linear function. The question asks us to find the y-intercept of the graph of y=f(x)y = f(x), which occurs when x=0x = 0.

Step-by-Step Solution:

Step 1: Analyze the table

From the table, we have the following values of g(x)g(x):

\hline x & g(x) \\ \hline -27 & 3 \\ -9 & 0 \\ 21 & 5 \\ \hline \end{array}$$ #### Step 2: Express $$f(x)$$ in terms of $$g(x)$$ We know that $$g(x) = \frac{f(x)}{x + 3}$$. This means: $$f(x) = g(x) \cdot (x + 3)$$ Now, we'll use this relationship to find $$f(x)$$ for each of the given values of $$x$$. #### Step 3: Calculate $$f(x)$$ at the given points 1. For $$x = -27$$: $$g(-27) = 3 \quad \text{so} \quad f(-27) = 3 \cdot (-27 + 3) = 3 \cdot (-24) = -72$$ 2. For $$x = -9$$: $$g(-9) = 0 \quad \text{so} \quad f(-9) = 0 \cdot (-9 + 3) = 0$$ 3. For $$x = 21$$: $$g(21) = 5 \quad \text{so} \quad f(21) = 5 \cdot (21 + 3) = 5 \cdot 24 = 120$$ Now we have the points $$(-27, -72)$$, $$(-9, 0)$$, and $$(21, 120)$$ for the linear function $$f(x)$$. #### Step 4: Find the equation of the linear function $$f(x)$$ The general form of a linear equation is: $$f(x) = mx + b$$ where $$m$$ is the slope, and $$b$$ is the y-intercept. We can find the slope $$m$$ using the two points $$(-9, 0)$$ and $$(21, 120)$$: $$m = \frac{120 - 0}{21 - (-9)} = \frac{120}{30} = 4$$ So the equation becomes: $$f(x) = 4x + b$$ #### Step 5: Solve for the y-intercept $$b$$ We can substitute one of the points, say $$(-9, 0)$$, into the equation $$f(x) = 4x + b$$ to solve for $$b$$: $$0 = 4(-9) + b \quad \Rightarrow \quad 0 = -36 + b \quad \Rightarrow \quad b = 36$$ #### Step 6: Conclusion The y-intercept of the graph of $$y = f(x)$$ is $$b = 36$$, so the correct answer is: $$\boxed{A: (0, 36)}$$ --- Would you like further details on any part of the solution? Here are five related questions to expand your understanding: 1. How do you find the equation of a line given two points? 2. What is the significance of the slope in a linear equation? 3. Can the method used here be applied to non-linear functions? 4. How do transformations of functions affect the intercepts? 5. What is the difference between the x-intercept and y-intercept in a graph? **Tip:** Always double-check the function given in a question. Often, understanding the relationship between different variables is key to solving a problem efficiently.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Functions
Intercepts
Algebraic Equations

Formulas

g(x) = f(x) / (x + 3)
Slope formula: m = (y2 - y1) / (x2 - x1)
Linear equation: y = mx + b

Theorems

Slope-Intercept Form

Suitable Grade Level

Grades 10-12

Related Recommendation

Find the y-Intercept of a Linear Function Using Given Values

Comparing Y-Intercepts of Quadratic Functions f(x) and h(x)

Find the y-intercept of y = (3x - 3) / (x + 1)

Find the y-intercept of the linear function f(x) = -(2/9)x + (1/3)

Find the y-intercept of g(x) after translating the parent function f(x) = x by 3 units upward