Exponential Functions Quiz
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Questions and Answers

What is the base of an exponential function in the form f(x)= ab^cx?

  • x
  • c
  • b (correct)
  • a
  • What is an exponential function?

    a function of the form f(x)= ab^cx where a, b, and c are real numbers, b > 0, b ≠ 1

    In an exponential function of the form f(x)= ab^cx, the number b is called the ____.

    base

    What are the restrictions on the value of b in an exponential function?

    <p>It must be a positive number and not equal to one.</p> Signup and view all the answers

    No matter what the base, an exponential function of the form f(x)=b^x always goes through the point ___.

    <p>(0, 1)</p> Signup and view all the answers

    An exponential function of the form f(x)=b^x always goes through the point ( ___, b), where b is the base.

    <p>1</p> Signup and view all the answers

    If f(x)=3^x and g(x)=7^x, then the graph of f(x) will be ___ the graph of g(x) when x > 0.

    <p>below</p> Signup and view all the answers

    If f(x)=0.5^x and g(x)=0.3^x, then the graph of f(x) will be _____ the graph of g(x) when x < 0.

    <p>above</p> Signup and view all the answers

    What is the mathematical constant e approximately equal to?

    <p>2.71828</p> Signup and view all the answers

    For an exponential function f(x)= ab^cx, changing the value for a will change the ____ to the value of a.

    <p>y-intercept</p> Signup and view all the answers

    For an exponential function f(x)= ab^cx, changing the value for c will change the ____ b to b^c.

    <p>base</p> Signup and view all the answers

    How can you find the inverse of a function?

    <p>By interchanging the numbers in each ordered pair of the function.</p> Signup and view all the answers

    The graph of the inverse of a function may be found by ___ over the line y=x.

    <p>reflecting</p> Signup and view all the answers

    The inverse of an exponential function tells us the ___ to which the base of an exponential function is raised to give us the value x.

    <p>exponent</p> Signup and view all the answers

    What is a common logarithm?

    <p>Logarithm with base 10, indicated by log x.</p> Signup and view all the answers

    What is a logarithm?

    <p>The logarithm <em>logb x</em> for a base <em>b</em> and a number <em>x</em> is defined to be the number <em>y</em>, so that <em>b^y=x</em>.</p> Signup and view all the answers

    What is a logarithmic function?

    <p>A function of the form <em>f(x)= logb^x</em>.</p> Signup and view all the answers

    What is a natural logarithm?

    <p>Logarithm with base <em>e</em>, indicated by <em>lnx</em>.</p> Signup and view all the answers

    The number, y so that b^y=x is called the ____ of x.

    <p>logarithm</p> Signup and view all the answers

    The base of the logarithm function cannot be equal to ________ and must be _____________.

    <p>one, positive</p> Signup and view all the answers

    The greater the base, the closer the graph is to the ___ when x>1.

    <p>x-axis</p> Signup and view all the answers

    To shift the graph of f(x)= a+clogb(dx+g) horizontally, change the parameter ____.

    <p>g</p> Signup and view all the answers

    To shift the graph of f(x)=a+clogb(dx+g) vertically, change parameters ___ or ___.

    <p>d, a</p> Signup and view all the answers

    To stretch the graph of f(x)=a+clogb(dx+g) change the parameter _____.

    <p>c</p> Signup and view all the answers

    Study Notes

    Exponential Functions

    • Exponential functions take the form f(x)= ab^cx, where a, b, and c are real numbers, with b being positive and not equal to 1.
    • The base of an exponential function, denoted as b, is fundamental to the function's characteristics.
    • Exponential functions always pass through the point (0, 1), regardless of the base.
    • For any base b, an exponential function will pass through the point (1, b).

    Characteristics of Exponential Graphs

    • If two exponential functions, such as f(x)=3^x and g(x)=7^x, are compared, f(x) will be below g(x) for x > 0.
    • Conversely, if f(x)=0.5^x and g(x)=0.3^x, then f(x) will be above g(x) when x < 0.

    Important Constants

    • The mathematical constant e is approximately equal to 2.71828 and is essential in exponential functions.

    Function Modifications

    • In the function f(x)= ab^cx:
      • Changing the value of a affects the y-intercept, moving it to the value of a.
      • Altering c changes the base to b^c.

    Inverse Functions

    • To find the inverse of a function, interchange the numbers in each ordered pair.
    • The graph of an inverse function can be derived by reflecting the original graph over the line y=x.
    • The inverse of an exponential function reveals the exponent needed to raise the base to achieve a given value, x.

    Logarithmic Functions

    • A common logarithm has a base of 10 and is denoted as log x.
    • The logarithm logb x is defined such that b^y = x, where y represents the logarithm of x with base b.
    • Logarithmic functions are expressed as f(x)= logb^x.
    • The natural logarithm, noted as ln x, uses base e.

    Restrictions and Graph Behavior

    • The base of a logarithm cannot be 1 and must remain positive.
    • When the base increases, the graph of the function approaches the x-axis as x > 1.

    Graph Shifting and Stretching

    • Horizontal shifts in the graph of the function f(x)=a+clogb(dx+g) occur when changing the parameter g.
    • Vertical shifts are made by altering either parameter d or a.
    • To stretch the graph vertically, change the parameter c.

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    Description

    Test your knowledge of exponential functions, including their characteristics and graphical representations. Understand the significance of the constants involved, such as base 'b' and the constant 'e'. Prepare to explore how modifications to the function affect its graph and intercepts.

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