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Questions and Answers
Given an absolute value function $f(x) = a|x - h| + k$, how does changing the value of 'a' affect the graph?
Given an absolute value function $f(x) = a|x - h| + k$, how does changing the value of 'a' affect the graph?
- It vertically stretches or compresses the graph and can reflect it over the x-axis. (correct)
- It shifts the graph vertically.
- It shifts the graph horizontally.
- It reflects the graph over the y-axis.
Consider the absolute value function $f(x) = |x|$. How does the graph of $g(x) = |x + c|$ compare if $c$ is a positive constant?
Consider the absolute value function $f(x) = |x|$. How does the graph of $g(x) = |x + c|$ compare if $c$ is a positive constant?
- The graph shifts 'c' units to the right.
- The graph shifts 'c' units downwards.
- The graph shifts 'c' units upwards.
- The graph shifts 'c' units to the left. (correct)
How does the domain of a quadratic function, represented by $f(x) = ax^2 + bx + c$, differ from the domain of a rational function, such as $g(x) = \frac{1}{x}$?
How does the domain of a quadratic function, represented by $f(x) = ax^2 + bx + c$, differ from the domain of a rational function, such as $g(x) = \frac{1}{x}$?
- Both quadratic and rational functions have the same domain.
- Quadratic functions have a domain of all real numbers, while rational functions may have restrictions where the denominator is zero. (correct)
- Rational functions always have a domain of all real numbers, while quadratic functions may have restrictions.
- Quadratic functions have restricted domains while rational functions do not.
Given the quadratic function $f(x) = a(x - h)^2 + k$, how would you determine the vertex of the parabola?
Given the quadratic function $f(x) = a(x - h)^2 + k$, how would you determine the vertex of the parabola?
What is the impact of a negative leading coefficient on a quadratic function, such as in $f(x) = -ax^2 + bx + c$ where $a > 0$?
What is the impact of a negative leading coefficient on a quadratic function, such as in $f(x) = -ax^2 + bx + c$ where $a > 0$?
How does changing the 'h' value in the vertex form of a quadratic equation, $f(x) = a(x - h)^2 + k$, transform the graph?
How does changing the 'h' value in the vertex form of a quadratic equation, $f(x) = a(x - h)^2 + k$, transform the graph?
If a quadratic function's vertex is at $(2, -3)$ and it opens upwards, which statement is true about its range?
If a quadratic function's vertex is at $(2, -3)$ and it opens upwards, which statement is true about its range?
Consider the inequality $y > |x - a| + b$. How does changing the value of 'b' affect the solution set?
Consider the inequality $y > |x - a| + b$. How does changing the value of 'b' affect the solution set?
Which transformation is applied to the parent function $f(x) = x^2$ to obtain the function $g(x) = (x - 3)^2 + 5$?
Which transformation is applied to the parent function $f(x) = x^2$ to obtain the function $g(x) = (x - 3)^2 + 5$?
If a graph of an absolute value function is vertically compressed by a factor of 2 and reflected over the x-axis, what transformation equation represents this?
If a graph of an absolute value function is vertically compressed by a factor of 2 and reflected over the x-axis, what transformation equation represents this?
Given the function $f(x) = a|x - h| + k$, what effect does changing the value of h have on the graph of the function..?
Given the function $f(x) = a|x - h| + k$, what effect does changing the value of h have on the graph of the function..?
How does the graph of $f(x) = |x|$ differ from the graph of of $g(x) = -|x|$?
How does the graph of $f(x) = |x|$ differ from the graph of of $g(x) = -|x|$?
Compare $f(x) = x^2$ and $f(x) = 3x^2$. What is true of the absolute value of their slopes as x increases?
Compare $f(x) = x^2$ and $f(x) = 3x^2$. What is true of the absolute value of their slopes as x increases?
Which transformations are present in $f(x) = (x+3)^2 - 5$?
Which transformations are present in $f(x) = (x+3)^2 - 5$?
How does the domain of an absolute value function $f(x) = |x|$ differ from the domain of $g(x) = \frac{1}{x}$
How does the domain of an absolute value function $f(x) = |x|$ differ from the domain of $g(x) = \frac{1}{x}$
If the graph of $f(x) = x^2$ is shifted down 3 and to the right 5, what function represents this transformation?
If the graph of $f(x) = x^2$ is shifted down 3 and to the right 5, what function represents this transformation?
The axis of symmetry for a paraobla is $x=3$ and passes through the point $(5,6)$. What is the y coordinate of the vertex?
The axis of symmetry for a paraobla is $x=3$ and passes through the point $(5,6)$. What is the y coordinate of the vertex?
How does adding a constant affect the range of a function?
How does adding a constant affect the range of a function?
A quadratic has a vertex of (3,4) and passes through (2,0). What other x-intercept exists?
A quadratic has a vertex of (3,4) and passes through (2,0). What other x-intercept exists?
What transformations are performed to the parent function $f(x) = |x|$ to obtain $g(x) = -2|x+3|-4$?
What transformations are performed to the parent function $f(x) = |x|$ to obtain $g(x) = -2|x+3|-4$?
Flashcards
What is the vertex?
What is the vertex?
The highest or lowest point on a graph; for absolute value functions, it's where the graph changes direction.
What is the domain?
What is the domain?
Represents all possible input values (x-values) for which the function is defined.
What is the range?
What is the range?
Represents all possible output values (y-values) of a function.
What is Vertical Stretch/Compression?
What is Vertical Stretch/Compression?
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What is horizontal shift?
What is horizontal shift?
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What is vertical shift?
What is vertical shift?
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What is Reflection in x-axis?
What is Reflection in x-axis?
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What is Axis of Symmetry?
What is Axis of Symmetry?
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What is reflection across the x-axis (quadratics)?
What is reflection across the x-axis (quadratics)?
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What is Parent Function?
What is Parent Function?
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Study Notes
- Domain and range are fundamental concepts for describing functions
- The domain is the set of all possible input values (x-values) for which a function is defined
- The range is the set of all possible output values (y-values) that the function can produce
Absolute Value Functions
- The absolute value of a number is its distance from zero, always non-negative
- The general form of an absolute value function is f(x) = a|x - h| + k, where (h, k) is the vertex
- The vertex represents the minimum or maximum point of the function
Transformations of Absolute Value Functions
- Vertical Stretch: If |a| > 1, the graph is stretched vertically, making it narrower
- Vertical Compression: If 0 < |a| < 1, the graph is compressed vertically, making it wider
- Reflection: If a < 0, the graph is reflected over the x-axis
- Horizontal Shift: The term (x - h) shifts the graph horizontally
- If h > 0, the graph shifts to the right by h units
- If h < 0, the graph shifts to the left by |h| units
- Vertical Shift: The term k shifts the graph vertically
- If k > 0, the graph shifts upward by k units
- If k < 0, the graph shifts downward by |k| units
- Vertex Form: f(x) = a|x - h| + k
Quadratic Functions
- A quadratic function is a polynomial function of degree 2, generally written as f(x) = ax² + bx + c
- The graph of a quadratic function is a parabola
- The vertex of a parabola is its minimum or maximum point
Axis of Symmetry
- The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetric halves
- Its equation is x = h, where h is the x-coordinate of the vertex
Vertex Form
- The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex
- From this form, the vertex (h, k) can be easily identified
- Convert between forms by completing the square
Transformations of Quadratic Functions
- Vertical Stretch/Compression: If |a| > 1, the graph is stretched vertically
- If 0 < |a| < 1, the graph is compressed vertically
- Reflection: If a < 0, the graph is reflected over the x-axis
- Horizontal shift: Described by h in f(x) = a(x - h)² + k
- Vertical shift: Described by k in f(x) = a(x - h)² + k
Inequalities
- When graphing inequalities, the area above or below the function is shaded
- Dashed line if the inequality is strict (< or >)
- Solid line if the inequality includes equality (≤ or ≥)
Piecewise Functions
- A piecewise function is defined by different expressions over different intervals of its domain
- Graph each piece separately over the specified interval
- Pay attention to the endpoints of each interval, using open or closed circles to indicate whether the endpoint is included
End Behavior
- End behavior describes what happens to the function values as x approaches positive or negative infinity
- For polynomials, the end behavior is determined by the leading term (term with the highest power of x)
Function Families
- Recognizing function families help in quickly understanding the type of graph
- Examples: Linear, absolute value, quadratic
Parent Functions
- The simplest form of a function family
- Linear: f(x) = x
- Absolute value: f(x) = |x|
- Quadratic: f(x) = x²
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