Math Problem Statement

The y-intercept of the exponential function y=6^x is 1.

Solution

The y-intercept of any function is the value of yy when x=0x = 0. Let's verify this for the exponential function y=6xy = 6^x:

  1. Substitute x=0x = 0 into the function: y=6x=60y = 6^x = 6^0

  2. Using the property of exponents a0=1a^0 = 1 (where a0a \neq 0): y=1y = 1

Thus, the y-intercept of y=6xy = 6^x is indeed 11.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

  1. What is the slope of the tangent line to y=6xy = 6^x at x=0x = 0?
  2. How does the base of the exponential function affect its y-intercept?
  3. What are the asymptotes of the function y=6xy = 6^x?
  4. How does the graph of y=6xy = 6^x compare to y=2xy = 2^x?
  5. If the function were y=6x+cy = 6^{x+c}, how would the y-intercept change?

Tip: The base of an exponential function always determines how steeply the graph grows or decays as xx increases or decreases.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Exponential Functions
Properties of Exponents
Graph Intercepts

Formulas

y = a^x
a^0 = 1

Theorems

Exponentiation Rules

Suitable Grade Level

Grades 9-10