Math Problem Statement

Which function has the same y-intercept as the function whose graph is shown?

Solution

We are asked to identify which function has the same y-intercept as the function whose graph is shown.

Steps to solve:

  1. Locate the y-intercept: The y-intercept is the point where the graph crosses the y-axis (when x=0x = 0). From the graph, it looks like the y-intercept is at y=2y = -2.

  2. Find the y-intercept for each function:

    • f(x)=2xf(x) = 2^x: When x=0x = 0, f(0)=20=1f(0) = 2^0 = 1. The y-intercept is 11.
    • f(x)=2x5+5x34x2+10f(x) = 2x^5 + 5x^3 - 4x^2 + 10: When x=0x = 0, f(0)=10f(0) = 10. The y-intercept is 1010.
    • f(x)=10x5+3xf(x) = -10x^5 + 3x: When x=0x = 0, f(0)=0f(0) = 0. The y-intercept is 00.
    • f(x)=x42f(x) = x^4 - 2: When x=0x = 0, f(0)=042=2f(0) = 0^4 - 2 = -2. The y-intercept is 2-2.

Conclusion:

The function that has the same y-intercept (2-2) as the graph shown is f(x)=x42f(x) = x^4 - 2.

Would you like more details or have any questions about the process?

Here are five related questions:

  1. How can you quickly find the y-intercept of any function?
  2. What is the significance of the y-intercept in real-world applications?
  3. How do different powers of xx affect the shape of a graph?
  4. Why does the exponential function 2x2^x have a positive y-intercept?
  5. How do you distinguish between different polynomial graphs?

Tip: Always substitute x=0x = 0 into a function to find the y-intercept!

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Math Problem Analysis

Mathematical Concepts

Y-intercepts
Functions
Graphs

Formulas

Substitute x = 0 into f(x) to find the y-intercept

Theorems

Y-intercept Theorem

Suitable Grade Level

Grades 9-11