Podcast
Questions and Answers
The graph of a quadratic function is a(n) _______.
The graph of a quadratic function is a(n) _______.
parabola
The function $f(x) = x^2$ is called the:
The function $f(x) = x^2$ is called the:
- Quadratic Parent Function (correct)
- Quadratic Regression
- Quadratic Child Function
- Quadratic Transform Function
To model the height of an object launched into the air $t$ seconds after it is launched, you can use the _______.
To model the height of an object launched into the air $t$ seconds after it is launched, you can use the _______.
vertical motion model
The _______ is $f(x) = ax^2 + bx + c.
The _______ is $f(x) = ax^2 + bx + c.
A(n) _______ is a method used to find a quadratic function that best fits a data set.
A(n) _______ is a method used to find a quadratic function that best fits a data set.
If the 'a' value in a quadratic function $f(x) = ax^2 + bx + c$ is negative, the parabola opens upwards.
If the 'a' value in a quadratic function $f(x) = ax^2 + bx + c$ is negative, the parabola opens upwards.
What is the vertex form of a quadratic function?
What is the vertex form of a quadratic function?
In the vertex form $f(x) = a(x - h)^2 + k$, what does 'h' represent?
In the vertex form $f(x) = a(x - h)^2 + k$, what does 'h' represent?
What does the 'c' value represent in the standard form of a quadratic equation, $f(x) = ax^2 + bx + c$?
What does the 'c' value represent in the standard form of a quadratic equation, $f(x) = ax^2 + bx + c$?
The axis of symmetry for a quadratic function in standard form is always the y-axis.
The axis of symmetry for a quadratic function in standard form is always the y-axis.
How do you determine the y-intercept of a quadratic function in standard form?
How do you determine the y-intercept of a quadratic function in standard form?
Given a quadratic function in vertex form, $f(x) = a(x - h)^2 + k$, what is the vertex of the parabola?
Given a quadratic function in vertex form, $f(x) = a(x - h)^2 + k$, what is the vertex of the parabola?
If a parabola opens downwards, it has a minimum value.
If a parabola opens downwards, it has a minimum value.
What is the formula for finding the axis of symmetry for a quadratic function in the standard form $f(x) = ax^2 + bx + c$?
What is the formula for finding the axis of symmetry for a quadratic function in the standard form $f(x) = ax^2 + bx + c$?
Match each quadratic function form with its defining characteristic:
Match each quadratic function form with its defining characteristic:
What is the first step in determining whether a table shows a linear, quadratic, or exponential function?
What is the first step in determining whether a table shows a linear, quadratic, or exponential function?
If the first differences in a table of values are constant, the function is quadratic.
If the first differences in a table of values are constant, the function is quadratic.
What type of function is represented if the ratios between consecutive y-values are constant?
What type of function is represented if the ratios between consecutive y-values are constant?
Alberto launches an emergency flare at an initial velocity of 64 ft/s from an initial height of 6 ft. Which of the following equations can be used to model the situation?
Alberto launches an emergency flare at an initial velocity of 64 ft/s from an initial height of 6 ft. Which of the following equations can be used to model the situation?
The _______ is $x = -b/2a$.
The _______ is $x = -b/2a$.
Flashcards
Parabola
Parabola
The graph of a quadratic function.
Quadratic Parent Function
Quadratic Parent Function
A basic quadratic function, f(x) = x².
Quadratic Regression
Quadratic Regression
A statistical method to find a quadratic function that best fits a data set.
Standard Form of a Quadratic Function
Standard Form of a Quadratic Function
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Vertex Form of a Quadratic Function
Vertex Form of a Quadratic Function
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Quadratic Functions in Vertex Form
Quadratic Functions in Vertex Form
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Quadratic Functions in Standard Form
Quadratic Functions in Standard Form
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Modeling With Quadratic Functions
Modeling With Quadratic Functions
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Linear, Exponential, and Quadratic Models
Linear, Exponential, and Quadratic Models
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Study Notes
Key Features of Quadratic Functions
- The graph of f(x) = ax² is a parabola with vertex (0, 0) and axis of symmetry x = 0.
- When a > 0, the parabola opens upward and the function has a minimum at the vertex.
- When a < 0, the parabola opens downward and the function has a maximum at the vertex.
- The graph of g(x) = -0.2x² opens downward and is wider than the graph of f(x) = x².
- For both graphs, the axis of symmetry is x = 0 and the vertex is (0, 0).
Modeling With Quadratic Functions
- Quadratic functions can model situations.
- A vertical motion model is a quadratic function.
- Alberto launches an emergency flare at an initial velocity of 64 ft/s from an initial height of 6 ft.
- Rescue team wants the flare must reach a height of 100 ft to be seen.
- 64 is substituted for v₀ and 6 for h₀ in the vertical motion model.
- The vertical motion equation is h(t) = -16t² + 64t + 6.
- The vertex (t, h(t)) can be found by finding t = -b/2a = -64/(2(-16)) = 2.
- The h(2) = -16(2)² + 64(2) + 6 = 70.
- Therefore the vertex is (2, 70).
- The flare will reach a maximum height of 70 ft, so Alberto’s launch is not successful.
Linear, Exponential, and Quadratic Models
- To determine which function best models a data set, analyze the differences and ratios between consecutive y-values when the differences in consecutive x-values are constant.
- Since the ratio between the y-values is constant, the function is exponential.
Quadratic Functions in Vertex Form
- The vertex form of a quadratic function is f(x) = a(x - h)² + k.
- The vertex of the graph is at (h, k) and the axis of symmetry is x = h.
- The function f(x) = (x + 1)² - 1 has a vertex at (-1, -1) and the axis of symmetry is x = -1.
Quadratic Functions in Standard Form
- The standard form of a quadratic function is f(x) = ax² + bx + c, where a ≠ 0.
- The y-intercept is c and the axis of symmetry, which is also the x-coordinate of the vertex, is x = -b/2a.
- For the function f(x) = 3x² - 6x + 2, the y-intercept is 2.
- The axis of symmetry can be found by calculating x = -b / 2a = -(-6) / 2(3) = 1.
- The y-coordinate of the vertex: f(1) = 3(1)² - 6(1) + 2 = -1.
- Plot the vertex (1, -1), identify the axis of symmetry and plot the y-intercept (0, 2).
- Reflect that point across the axis of symmetry.
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