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Questions and Answers
What is the domain of the function $f(x) = 2 - 0.4x$?
What is the domain of the function $f(x) = 2 - 0.4x$?
- Only integer values
- All negative numbers
- All real numbers (correct)
- All positive numbers
If $f(t) = 2t - t^2$, what does the graph of the function represent over time for a pie cooling?
If $f(t) = 2t - t^2$, what does the graph of the function represent over time for a pie cooling?
- A parabolic curve that rises then falls (correct)
- Consistent increase in temperature
- Straight line with no change in temperature
- Consistent decrease in temperature
How does the height of grass change over a four-week period as modeled by the homeowner's mowing schedule?
How does the height of grass change over a four-week period as modeled by the homeowner's mowing schedule?
- Exponential growth without interruption
- Fluctuating growth depending on mowing frequency (correct)
- Constant decrease in height
- Linear growth of height with no reductions
What does the function $H(t) = 3x - x^2$ suggest about height over time?
What does the function $H(t) = 3x - x^2$ suggest about height over time?
If an airplane travels 400 miles in one hour, what is its average speed?
If an airplane travels 400 miles in one hour, what is its average speed?
What effect does time $t$ have on the function $f(t) = 4 - t^2$ in relation to temperature changes?
What effect does time $t$ have on the function $f(t) = 4 - t^2$ in relation to temperature changes?
What is the output of the function f for the input x = 0?
What is the output of the function f for the input x = 0?
How does the function $G(x) = 3x - x^2$ behave as $x$ increases?
How does the function $G(x) = 3x - x^2$ behave as $x$ increases?
What is the value of f when x = 1?
What is the value of f when x = 1?
For x > 1, the function f(x) is defined as which of the following expressions?
For x > 1, the function f(x) is defined as which of the following expressions?
Given the function $t(x) = x^2 + 1$ for $x
eq 0$, how does this function behave for negative $x$ values?
Given the function $t(x) = x^2 + 1$ for $x eq 0$, how does this function behave for negative $x$ values?
What does the solid dot on the graph of f at the point (1,0) indicate?
What does the solid dot on the graph of f at the point (1,0) indicate?
What is the slope of the line representing f(x) for x < 1?
What is the slope of the line representing f(x) for x < 1?
Which statement is true about the value of f for any x such that |x| > 0?
Which statement is true about the value of f for any x such that |x| > 0?
What is the behavior of f(x) when x approaches 1 from the left?
What is the behavior of f(x) when x approaches 1 from the left?
The absolute value function a is defined as which of the following?
The absolute value function a is defined as which of the following?
What is the y-intercept of the linear function T?
What is the y-intercept of the linear function T?
What value of T corresponds to h = 1?
What value of T corresponds to h = 1?
What is the slope (m) of the linear function T?
What is the slope (m) of the linear function T?
How can the linear function T be expressed in terms of h?
How can the linear function T be expressed in terms of h?
If h increases by 1, what is the change in value of T?
If h increases by 1, what is the change in value of T?
What characterizes an even function?
What characterizes an even function?
Which of the following examples is an even function?
Which of the following examples is an even function?
What does the graph of an even function exhibit regarding its symmetry?
What does the graph of an even function exhibit regarding its symmetry?
In the function f(x) = -x^2, what can be concluded about its evenness?
In the function f(x) = -x^2, what can be concluded about its evenness?
Why are functions like step functions described as jumping?
Why are functions like step functions described as jumping?
Which of the following properties does NOT define an even function?
Which of the following properties does NOT define an even function?
What does the expression f(-x) indicate in relation to function analysis?
What does the expression f(-x) indicate in relation to function analysis?
How can the symmetry of an even function be visually confirmed?
How can the symmetry of an even function be visually confirmed?
What type of graph would the line segment joining the points (-1, -3) and (-5, 7) produce?
What type of graph would the line segment joining the points (-1, -3) and (-5, 7) produce?
If an even function includes the point (-5, 10), which of the following points must also be included?
If an even function includes the point (-5, 10), which of the following points must also be included?
For the even function graph, if the point (-5, 3) is included, what must be the corresponding point?
For the even function graph, if the point (-5, 3) is included, what must be the corresponding point?
What shape does the bottom half of the parabola given by the equation x = - (y - 1)^2 + 0 represent?
What shape does the bottom half of the parabola given by the equation x = - (y - 1)^2 + 0 represent?
How is the perimeter of a rectangle with a fixed area of 16 m² expressed in terms of its length?
How is the perimeter of a rectangle with a fixed area of 16 m² expressed in terms of its length?
If the area of the rectangle is fixed at 16 m², what expression provides the perimeter in relation to the sides?
If the area of the rectangle is fixed at 16 m², what expression provides the perimeter in relation to the sides?
Which of these best describes how to find the expression for the top half of the circle defined by the equation x^2 = - (y - 2)^2 + 4?
Which of these best describes how to find the expression for the top half of the circle defined by the equation x^2 = - (y - 2)^2 + 4?
What is the formula for the area of the rectangle with a fixed perimeter of 20 m expressed in terms of one side's length?
What is the formula for the area of the rectangle with a fixed perimeter of 20 m expressed in terms of one side's length?
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Study Notes
Function Domain
- The domain of a function defines all possible input values for which the function can produce a valid output.
- In the case of the function $f(x) = 2 - 0.4x$, the domain encompasses all real numbers because there are no restrictions on the input value $x$.
Function Graphs and Time
- The graph of a function can represent how a quantity changes over time.
- For example, the function $f(t) = 2t - t^2$ models the cooling of a pie, with $t$ representing time.
- The graph would depict the temperature change with time as the pie cools.
Function Behavior and Growth
- Functions can be used to model real-world scenarios like the growth of grass.
- The function $G(x) = 3x - x^2$ would describe the height of the grass over time, with $x$ representing the number of weeks.
- As $x$ increases, the function's behavior would show the grass growing, reaching a peak height, and then eventually decreasing.
Function Output and Input
- For a given function, the output is the value the function produces for a specific input value.
- For the function $f(x) = 2 - 0.4x$, when the input $x = 0$, the output $f(0) = 2$.
Function Values and Expressions
- The value of a function at a specific input is determined by substituting the value into the function's expression.
- For example, if $f(x) = 2 - 0.4x$, then $f(1) = 1.6$.
Function Behavior and Graphs
- The graph of a function can provide insights into its behavior.
- For example, the graph of the function $f(x)$ might show a change in its slope at a certain point, indicating a shift in its rate of change.
Absolute Value Functions
- An absolute value function takes the magnitude of a number, disregarding its sign.
- The absolute value function, denoted as $|x|$, gives the distance of $x$ from zero.
Linear Functions and Slope
- A linear function is represented by a straight line on a graph.
- Its slope (m) defines the steepness of the line, indicating the rate of change.
Function Symmetry
- Even functions exhibit symmetry about the y-axis.
- This means that for any input $x$, the function's output is the same as its output for $-x$.
Even Functions and Examples
- An even function is characterized by the property $f(x) = f(-x)$ for all values of $x$.
- For example, the function $f(x) = -x^2$ is an even function as $f(x) = f(-x)$ holds true for all values of $x$.
Function Graphs and Symmetry
- The graph of an even function exhibits symmetry about the y-axis.
- This means that the graph can be reflected across the y-axis and the resulting image will be identical to the original graph.
Function Analysis and Symmetry
- The expression $f(-x)$ is used to analyze the symmetry of a function.
- If $f(-x) = f(x)$, the function is even.
- If $f(-x) = -f(x)$, the function is odd.
Graphing Line Segments
- A line segment connecting two points on a graph is represented by a straight line.
- The slope of the line segment can be calculated using the formula (y2 - y1) / (x2 - x1).
Even Functions and Symmetry Properties
- Even functions have symmetry about the y-axis. If a point (a, b) exists on the graph of an even function, then the point (-a, b) also exists on the graph.
Parabola Shapes and Equations
- The equation x = - (y - 1)^2 + 0 defines a parabola that opens to the left.
- The bottom half of the parabola is represented by the part of the curve below the vertex.
Perimeter of Rectangles with Fixed Area
- If a rectangle has a fixed area, its perimeter can be expressed as a function of its length.
- For example, if the area of a rectangle is 16 m², the perimeter can be expressed as P = 2l + 32/l, where l is the length of the rectangle.
Finding the Top Half of a Circle
- The equation x^2 = - (y - 2)^2 + 4 defines a circle centered at (0, 2) with a radius of 2.
- To find the expression for the top half of the circle, solve for y.
Area of Rectangles with Fixed Perimeter
- If a rectangle has a fixed perimeter of 20 m, its area can be expressed as a function of one side's length.
- Let l represent the length of the rectangle, then the area can be expressed as A = 10l - l^2.
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