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Questions and Answers
A tank's water height is measured at different times. At t = 2 minutes, the height is 4 feet, and at t = 6 minutes, the height is 12 feet. What is the average rate of change of the water height?
A tank's water height is measured at different times. At t = 2 minutes, the height is 4 feet, and at t = 6 minutes, the height is 12 feet. What is the average rate of change of the water height?
- 1 ft/min
- 4 ft/min
- 2 ft/min (correct)
- 3 ft/min
For a linear function, the rate of change is constant, while for a quadratic function, the rate of change varies.
For a linear function, the rate of change is constant, while for a quadratic function, the rate of change varies.
True (A)
Given the function $f(x) = 2x^2$, calculate the average rate of change from $x = 0$ to $x = 3$.
Given the function $f(x) = 2x^2$, calculate the average rate of change from $x = 0$ to $x = 3$.
6
The end behavior of a polynomial function is determined by its ______.
The end behavior of a polynomial function is determined by its ______.
Match the following functions with their respective rates of change:
Match the following functions with their respective rates of change:
Which of the following statements accurately describes the end behavior of the polynomial function $f(x) = -3x^4 + 2x^2 - x$?
Which of the following statements accurately describes the end behavior of the polynomial function $f(x) = -3x^4 + 2x^2 - x$?
The polynomial equation $x^2 + 9 = 0$ has real roots.
The polynomial equation $x^2 + 9 = 0$ has real roots.
Identify the zero of the rational function $f(x) = \frac{x + 5}{x - 3}$.
Identify the zero of the rational function $f(x) = \frac{x + 5}{x - 3}$.
A ______ occurs in a rational function when a factor cancels from both the numerator and the denominator.
A ______ occurs in a rational function when a factor cancels from both the numerator and the denominator.
Which transformation is applied to the function $f(x)$ to obtain $f(x + 4)$?
Which transformation is applied to the function $f(x)$ to obtain $f(x + 4)$?
The function $f(x) = \frac{3x}{x-2}$ has a vertical asymptote at what value of x?
The function $f(x) = \frac{3x}{x-2}$ has a vertical asymptote at what value of x?
The rate of change is always constant for quadratic functions.
The rate of change is always constant for quadratic functions.
Determine the horizontal asymptote of the rational function: $f(x) = \frac{4x^2 + 1}{2x^2 - 3}$
Determine the horizontal asymptote of the rational function: $f(x) = \frac{4x^2 + 1}{2x^2 - 3}$
If $f(x) = x^3$, then $-f(x)$ represents a reflection over the ______-axis.
If $f(x) = x^3$, then $-f(x)$ represents a reflection over the ______-axis.
Given that a graph levels off as x increases, which type of function would be most appropriate to model this behavior?
Given that a graph levels off as x increases, which type of function would be most appropriate to model this behavior?
Which of the following indicates that two quantities change together and are related?
Which of the following indicates that two quantities change together and are related?
The vertical asymptote of a rational function occurs when both the numerator and the denominator equal zero at the same x-value.
The vertical asymptote of a rational function occurs when both the numerator and the denominator equal zero at the same x-value.
What is the degree of the polynomial function $f(x)=5x^4-3x^2+2x-1$?
What is the degree of the polynomial function $f(x)=5x^4-3x^2+2x-1$?
If $f(x)$ is shifted vertically upwards by $k$ units, the new function is represented as $f(x)$ ______ $k$.
If $f(x)$ is shifted vertically upwards by $k$ units, the new function is represented as $f(x)$ ______ $k$.
Match the function transformation with the correct description:
Match the function transformation with the correct description:
Flashcards
Function
Function
A relationship where one quantity changes in response to another; Represented by graphs, tables, or equations.
Average Rate of Change
Average Rate of Change
The ratio of the change in the output value to the change in the input value of a function.
Linear Function
Linear Function
A function where the rate of change is constant.
Quadratic Function
Quadratic Function
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Polynomial
Polynomial
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Zeros of a Function
Zeros of a Function
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End Behavior
End Behavior
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Rational Function
Rational Function
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Zero of a Rational Function
Zero of a Rational Function
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Vertical Asymptote
Vertical Asymptote
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Hole in a Rational Function
Hole in a Rational Function
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Transformation of Functions
Transformation of Functions
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Function Model Selection
Function Model Selection
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Function Model Construction
Function Model Construction
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Study Notes
- A function relates two quantities that change together and can be represented with graphs, tables, and equations.
- For example, the height of water increasing in a tank over time represents a function.
- The rate of change can be calculated; for instance, if height increases from 0 ft at t=0 min to 10 ft at t=5 min, the rate is 2 ft/min.
- Graphs and tables can be used to describe increasing or decreasing behavior.
Rates of Change
- The average rate of change is the ratio of the change in output to the change in input.
- The average rate of change for a function f(x) from x=a to x=b is (f(b) - f(a)) / (b - a).
- As an example, the average rate of f(x) = x² from x = 1 to x = 4 is 5.
Rates of Change in Linear and Quadratic Functions
- Linear functions have a constant rate of change.
- Quadratic functions have a rate of change that varies.
- The slope is always constant in a linear function, such as f(x) = 3x + 2 where the slope is 3.
- The rate of change increases as x increases in a quadratic function like f(x) = x².
Polynomial Functions and Rates of Change
- A polynomial is an expression, such as f(x) = x³ - 2x + 1.
- Polynomials can be analyzed by degree (highest exponent) and end behavior (what happens as x approaches ±∞).
- The graph of f(x) = x³ - 2x has a degree of 3 and falls to the left and rises to the right.
Polynomial Functions and Complex Zeros
- Zeros are the solutions when f(x) = 0.
- Factoring reveals real and complex solutions.
- The solutions to x² + 4 = 0 include imaginary numbers: x = ±2i.
Polynomial Functions and End Behavior
- The leading term describes end behavior.
- If f(x) = 4x⁵ - x² + 3, the leading term 4x⁵ indicates that as x → ∞, f(x) → ∞, and as x → -∞, f(x) → -∞.
Rational Functions and End Behavior
- A rational function is a ratio of two polynomials.
- End behavior depends on the degrees of the numerator and denominator.
- If f(x) = (2x² + 1)/(x² + 4), there’s a horizontal asymptote at y = 2 because the degrees are equal.
Rational Functions and Zeros
- Zeros occur when the numerator = 0, but the denominator ≠ 0.
- For f(x) = (x - 3)/(x + 2), the zero is at x = 3.
Rational Functions and Vertical Asymptotes
- Vertical asymptotes occur when the denominator = 0 and numerator ≠ 0.
- For f(x) = (x - 1)/(x - 2), a vertical asymptote exists at x = 2.
Rational Functions and Holes
- Holes occur when numerator and denominator share a common factor that cancels.
- The function f(x) = (x - 2)(x + 1)/(x - 2) has a hole at x = 2.
Equivalent Representations of Polynomial and Rational Expressions
- The goal is to factor and simplify expressions.
- For example, (x² - 4)/(x - 2) simplifies to x + 2, where x ≠ 2.
Transformations of Functions
- Transformations include shifts, stretches, and reflections of functions.
- For f(x) = x², f(x) + 3 shifts the function up 3 units.
- f(x - 2) shifts the function right 2 units.
- -f(x) reflects the function over the x-axis.
Function Model Selection and Assumption Articulation
- Choose models that are appropriate based on graphs, context, and data.
- A rational model may be appropriate if a graph levels off as x increases.
Function Model Construction and Application
- Build a model from data or a context, such as revenue, population, or physics.
- Given data points for a falling object, a quadratic regression can be used to model height over time.
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