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Questions and Answers
What must be true of a quadratic function whose vertex is the same as its y-intercept?
What must be true of a quadratic function whose vertex is the same as its y-intercept?
The axis of symmetry for the function is x = 0.
What is the vertex type for the function f(x) = -x² - 2x - 1?
What is the vertex type for the function f(x) = -x² - 2x - 1?
maximum value
While teaching at the University of Pisa in 1589, Galileo tested Aristotle's laws of falling objects by dropping large metal projectiles off the _______________.
While teaching at the University of Pisa in 1589, Galileo tested Aristotle's laws of falling objects by dropping large metal projectiles off the _______________.
leaning tower of pisa
What did Galileo conclude about heavier and lighter objects dropped from the same height?
What did Galileo conclude about heavier and lighter objects dropped from the same height?
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What type of graph represents a projectile?
What type of graph represents a projectile?
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What is the maximum height of a projectile with an initial velocity of 48 feet per second launched from a 190-foot tall building?
What is the maximum height of a projectile with an initial velocity of 48 feet per second launched from a 190-foot tall building?
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Approximately when will a projectile launched with an initial velocity of 48 feet per second from a 190-foot tall building hit the ground?
Approximately when will a projectile launched with an initial velocity of 48 feet per second from a 190-foot tall building hit the ground?
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What was the maximum height reached by a cannonball launched from the ground with an initial velocity of 144 feet per second after 4.5 seconds?
What was the maximum height reached by a cannonball launched from the ground with an initial velocity of 144 feet per second after 4.5 seconds?
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Which function has an axis of symmetry at x = 3?
Which function has an axis of symmetry at x = 3?
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What coordinates could describe a graph with its maximum located at (-4, 2)?
What coordinates could describe a graph with its maximum located at (-4, 2)?
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What will be true for all functions of the form f(x) = ax² + bx + c when b = 0?
What will be true for all functions of the form f(x) = ax² + bx + c when b = 0?
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What is the y-intercept of the graph of the function f(x) = x² + 3x + 5?
What is the y-intercept of the graph of the function f(x) = x² + 3x + 5?
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If the axis of symmetry for the graph of f(x) = 3x² + bx + 4 is x = 3/2, what is the value of b?
If the axis of symmetry for the graph of f(x) = 3x² + bx + 4 is x = 3/2, what is the value of b?
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Over which interval is the graph of f(x) = -x² + 3x + 8 increasing?
Over which interval is the graph of f(x) = -x² + 3x + 8 increasing?
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The a value of a function in the form f(x) = ax² + bx + c is negative, which means the vertex is a maximum.
The a value of a function in the form f(x) = ax² + bx + c is negative, which means the vertex is a maximum.
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What are the coordinates of the vertex for the function f(x) = -x² - 10x + 16 if the axis of symmetry is x = -5?
What are the coordinates of the vertex for the function f(x) = -x² - 10x + 16 if the axis of symmetry is x = -5?
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Which statements about the graph of the function f(x) = -x² - 4x + 2 are true? Select three options.
Which statements about the graph of the function f(x) = -x² - 4x + 2 are true? Select three options.
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What is the x-intercept of the graph of the function f(x) = x² − 16x + 64?
What is the x-intercept of the graph of the function f(x) = x² − 16x + 64?
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Study Notes
Quadratic Functions Overview
- A quadratic function with its vertex as the y-intercept has an axis of symmetry at x = 0.
- The equation f(x) = -x² - 2x - 1 indicates a downward-opening parabola with a maximum vertex and a range of all real numbers less than or equal to zero.
Historical Context
- Galileo Galilei tested Aristotle's laws of motion in 1589 by dropping projectiles from the Leaning Tower of Pisa, concluding heavy and light objects fall at approximately the same time.
Projectile Motion
- The path of a projectile, modeled by h(t) = -16t² + 48t + 190, launched from a height of 190 feet with an initial velocity of 48 feet per second, reaches a maximum height of 226 feet.
- The same projectile hits the ground approximately 5.3 seconds after launch.
Cannonball Launch Example
- A cannonball launched from the ground with an initial velocity of 144 feet per second achieves a maximum height of 324 feet in 4.5 seconds.
Axis of Symmetry and Vertex
- The function f(x) = x² - 6x - 1 has an axis of symmetry at x = 3.
- For the graph with a maximum vertex at (-4, 2), there are specific graphs that could represent the function.
Function Characteristics
- When b = 0 in the quadratic function f(x) = ax² + bx + c, the y-intercept corresponds with the vertex.
- The y-intercept of f(x) = x² + 3x + 5 is at (0, 5).
- If the axis of symmetry is x = 3/2 for f(x) = 3x² + bx + 4, the value of b is determined to be -9.
Increasing and Decreasing Intervals
- The graph of f(x) = -x² + 3x + 8 is increasing over the interval (-∞, 1.5).
- A negative 'a' value in the equation f(x) = ax² + bx + c indicates that the vertex represents a maximum point.
Vertex Coordinates
- The vertex of f(x) = -x² - 10x + 16, with an axis of symmetry at x = -5, is at the coordinates (-5, 41).
Function Analysis
- For f(x) = -x² - 4x + 2, valid statements include:
- The range is {y | y ≤ 6}.
- The function is increasing over the interval (-∞, -2).
- The function has a positive y-intercept.
X-Intercept Calculation
- The x-intercept of the quadratic function f(x) = x² - 16x + 64 is (8, 0).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge of quadratic functions with these flashcards that cover essential concepts such as vertex and y-intercepts. Designed for Algebra 1 students, this quiz will help reinforce your understanding of standard form and properties of quadratic equations.
