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Questions and Answers
Given the quadratic function $f(x) = (x - 5)^2 + 3$, what is the range of this function?
Given the quadratic function $f(x) = (x - 5)^2 + 3$, what is the range of this function?
- $y \ge 3$ (correct)
- All real numbers
- $y \le 3$
- $x \ge 5$
For the absolute value function $g(x) = -|x + 4| - 2$, what is the maximum value of the function?
For the absolute value function $g(x) = -|x + 4| - 2$, what is the maximum value of the function?
- -4
- -2 (correct)
- No maximum value
- 2
Consider the square root function $h(x) = \sqrt{x - 3} - 4$. What is the domain of this function?
Consider the square root function $h(x) = \sqrt{x - 3} - 4$. What is the domain of this function?
- $x \le 3$
- All real numbers
- $x \ge 3$ (correct)
- $x \ge 4$
Which statement accurately describes the relationship between a function's leading term and its end behavior?
Which statement accurately describes the relationship between a function's leading term and its end behavior?
Given the investment function $f(x) = 5000(1.08)^x$, what is the most appropriate domain for evaluating the investment's value?
Given the investment function $f(x) = 5000(1.08)^x$, what is the most appropriate domain for evaluating the investment's value?
If an investment is modeled by the function $f(x) = 5000(1.08)^x$, where $x$ is the number of years, what does the value 5000 represent?
If an investment is modeled by the function $f(x) = 5000(1.08)^x$, where $x$ is the number of years, what does the value 5000 represent?
What is the range of the investment function $f(x) = 5000(1.08)^x$, considering only realistic investment values?
What is the range of the investment function $f(x) = 5000(1.08)^x$, considering only realistic investment values?
In the context of the investment function $f(x) = 5000(1.08)^x$, why might the end behavior predicted by this function be considered unrealistic over a very long term?
In the context of the investment function $f(x) = 5000(1.08)^x$, why might the end behavior predicted by this function be considered unrealistic over a very long term?
How does the graph of $f(x) = (x - 5)^2 + 3$ differ from the graph of $f(x) = x^2$?
How does the graph of $f(x) = (x - 5)^2 + 3$ differ from the graph of $f(x) = x^2$?
Given the function $g(x) = -|x + 4| - 2$, how does the negative sign affect the graph of the absolute value function?
Given the function $g(x) = -|x + 4| - 2$, how does the negative sign affect the graph of the absolute value function?
Flashcards
Domain of a function
Domain of a function
The set of all possible input values (x-values) for a function.
Range of a function
Range of a function
The set of all possible output values (y-values) for a function.
Maximum value
Maximum value
A point where a function reaches its highest value.
Minimum value
Minimum value
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Axis of symmetry
Axis of symmetry
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End Behavior
End Behavior
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Study Notes
Analyzing Functions Graphically
- Functions are graphed to state the domain and range.
Quadratic Function
- The quadratic function is f(x) = (x-5)^2 + 3.
- The domain is all real numbers.
- The range is y ≥ 3.
Absolute Value Function
- The absolute value function is g(x) = -|x+4| - 2.
- The domain is all real numbers.
- The range is y ≥ -2.
Square Root Function
- The square root function is h(x) = √(x-3) - 4.
- The domain is x ≥ 3.
- The range is y ≥ -4.
Minimum, Maximum Values, and Axis of Symmetry
- The minimum and maximum values and the equation for the axis of symmetry can be determined from the graph of each function.
Function 1
- It has a minimum value of 3.
- It has no maximum value.
- The axis of symmetry is 5.
Function 2
- It has no minimum value.
- It has a maximum value of -2.
- The axis of symmetry is -4.
Function 3
- It has a minimum value of -4.
- It has no maximum value.
- It has no axis of symmetry.
End Behavior of a Graph
- The end behavior of a graph can be determined without graphing by looking at the leading term.
Investment Function
- The function f(x) = 5,000(1.08)^x models an investment's value earning 8% per year for x years.
- The domain includes all whole numbers.
- The range is y ≥ 5,000.
- The end behavior is unrealistic because rates change over the years.
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