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Questions and Answers
If a function has an asymptote, what happens to the graph as you trace along it towards the left or right?
If a function has an asymptote, what happens to the graph as you trace along it towards the left or right?
- It moves away from the asymptote.
- It intersects the asymptote.
- It gets closer and closer to the asymptote. (correct)
- It oscillates around the asymptote.
The graph of a function can only have vertical asymptotes.
The graph of a function can only have vertical asymptotes.
False (B)
In the context of graphing functions, what is an asymptote?
In the context of graphing functions, what is an asymptote?
A line that a curve approaches but never touches.
A mathematically clear and complete definition of an __________ requires some ideas from calculus.
A mathematically clear and complete definition of an __________ requires some ideas from calculus.
What is the primary purpose of creating a table of x and y values when investigating a function?
What is the primary purpose of creating a table of x and y values when investigating a function?
When creating an x → y table to investigate a function, choosing x-values very close to a particular value is not useful for understanding the function's behavior near that point.
When creating an x → y table to investigate a function, choosing x-values very close to a particular value is not useful for understanding the function's behavior near that point.
Why is it important to choose x-values on both sides of h when investigating a function around x = h?
Why is it important to choose x-values on both sides of h when investigating a function around x = h?
Match the description with the corresponding concept related to function investigation:
Match the description with the corresponding concept related to function investigation:
A graph has a(n) __________ asymptote if, as you choose coordinates closer and closer to a certain value, the y-coordinate gets farther away from zero.
A graph has a(n) __________ asymptote if, as you choose coordinates closer and closer to a certain value, the y-coordinate gets farther away from zero.
Summary statements about a function should be justified with which of the following?
Summary statements about a function should be justified with which of the following?
What are the key features of quadratic functions you can determine using the PHET quadratic simulator?
What are the key features of quadratic functions you can determine using the PHET quadratic simulator?
Changing the coefficients of a quadratic function does not affect the graph of the function.
Changing the coefficients of a quadratic function does not affect the graph of the function.
Which geometric transformation is represented by flipping a graph over the x-axis?
Which geometric transformation is represented by flipping a graph over the x-axis?
A __________ involves shifting a graph without changing its shape or size.
A __________ involves shifting a graph without changing its shape or size.
What is the effect of the '+25' in the function $f(x) = \frac{1}{x} + 25$ on the graph?
What is the effect of the '+25' in the function $f(x) = \frac{1}{x} + 25$ on the graph?
Match the following terms with their description:
Match the following terms with their description:
Which of the following is NOT a reliable way to determine if a relationship is quadratic?
Which of the following is NOT a reliable way to determine if a relationship is quadratic?
Changing the coefficient 'c' in the quadratic equation $y = ax^2 + bx + c$ affects the width of the parabola.
Changing the coefficient 'c' in the quadratic equation $y = ax^2 + bx + c$ affects the width of the parabola.
In the standard form of a quadratic equation, what does the vertex of the parabola represent?
In the standard form of a quadratic equation, what does the vertex of the parabola represent?
The axis of symmetry is a vertical line that passes through the ______ of a parabola.
The axis of symmetry is a vertical line that passes through the ______ of a parabola.
How can you determine whether the vertex of a parabola is a maximum or a minimum?
How can you determine whether the vertex of a parabola is a maximum or a minimum?
Match each quadratic equation transformation to its effect on the graph of the parent function $y = x^2$:
Match each quadratic equation transformation to its effect on the graph of the parent function $y = x^2$:
The roots (or zeros) of a quadratic equation correspond to which feature on the graph of the parabola?
The roots (or zeros) of a quadratic equation correspond to which feature on the graph of the parabola?
Describe how changing the 'a' coefficient in the quadratic equation $y=ax^2$ transforms the graph of the parent function $y=x^2$.
Describe how changing the 'a' coefficient in the quadratic equation $y=ax^2$ transforms the graph of the parent function $y=x^2$.
The vertex form of a quadratic equation is given by $y = a(x-h)^2 + k$, where (h, k) represents the coordinates of the ______.
The vertex form of a quadratic equation is given by $y = a(x-h)^2 + k$, where (h, k) represents the coordinates of the ______.
Which of the following transformations does the 'a' value in the vertex form equation $y = a(x - h)^2 + k$ NOT directly control?
Which of the following transformations does the 'a' value in the vertex form equation $y = a(x - h)^2 + k$ NOT directly control?
In the quadratic equation $y = a(x - h)^2 + k$, the 'h' value directly indicates the y-coordinate of the vertex.
In the quadratic equation $y = a(x - h)^2 + k$, the 'h' value directly indicates the y-coordinate of the vertex.
Given the quadratic equation $y = (x + 3)^2 - 2$, which of the following statements is true?
Given the quadratic equation $y = (x + 3)^2 - 2$, which of the following statements is true?
Describe the transformation of the parent function $y = x^2$ to the function $y = 2(x - 1)^2 + 3$.
Describe the transformation of the parent function $y = x^2$ to the function $y = 2(x - 1)^2 + 3$.
If a parabola has a vertex at (2, -5) and opens downwards, which of the following equations could represent it?
If a parabola has a vertex at (2, -5) and opens downwards, which of the following equations could represent it?
Match the effect on the graph of a quadratic function with the corresponding change in its vertex form $y = a(x - h)^2 + k$:
Match the effect on the graph of a quadratic function with the corresponding change in its vertex form $y = a(x - h)^2 + k$:
If the 'a' value in the vertex form $y = a(x-h)^2 + k$ is negative, the parabola opens ______.
If the 'a' value in the vertex form $y = a(x-h)^2 + k$ is negative, the parabola opens ______.
Given a quadratic equation in standard form, what is the first step to finding the vertex?
Given a quadratic equation in standard form, what is the first step to finding the vertex?
The y-intercept can be directly determined from the factored form of a quadratic equation.
The y-intercept can be directly determined from the factored form of a quadratic equation.
Describe how changing the 'a' value in the standard form of a quadratic equation, $y = ax^2 + bx + c$, affects the shape of the parabola.
Describe how changing the 'a' value in the standard form of a quadratic equation, $y = ax^2 + bx + c$, affects the shape of the parabola.
In the graphing form of a quadratic equation, $y = a(x - h)^2 + k$, the vertex of the parabola is represented by the point (__ , __).
In the graphing form of a quadratic equation, $y = a(x - h)^2 + k$, the vertex of the parabola is represented by the point (__ , __).
Match each form of a quadratic equation with its most readily apparent feature:
Match each form of a quadratic equation with its most readily apparent feature:
Which transformation does the equation $y = -x^2$ represent when compared to the parent function $y = x^2$?
Which transformation does the equation $y = -x^2$ represent when compared to the parent function $y = x^2$?
Explain the difference between solutions written in exact form versus approximate form for a quadratic equation.
Explain the difference between solutions written in exact form versus approximate form for a quadratic equation.
To make a parabola narrower compared to its parent function while keeping the vertex the same, you should ______ the absolute value of the 'a' coefficient in the equation.
To make a parabola narrower compared to its parent function while keeping the vertex the same, you should ______ the absolute value of the 'a' coefficient in the equation.
When modeling the path of a jackrabbit jumping over a fence with a parabola, which placement of the x- and y-axes would generally simplify the equation-finding process?
When modeling the path of a jackrabbit jumping over a fence with a parabola, which placement of the x- and y-axes would generally simplify the equation-finding process?
If a parabola models the path of a soccer ball kicked from the ground, with a maximum height of 100 feet and a horizontal distance of 150 feet, the vertex of the parabola will be at (75, 100).
If a parabola models the path of a soccer ball kicked from the ground, with a maximum height of 100 feet and a horizontal distance of 150 feet, the vertex of the parabola will be at (75, 100).
A U-Dip at a skateboard park has a cross-section modeled by a parabola. If the dip is 15 feet below ground and 40 feet wide at ground level, what is the distance from the lowest point of the U-Dip to one of the points where the U-Dip meets ground level?
A U-Dip at a skateboard park has a cross-section modeled by a parabola. If the dip is 15 feet below ground and 40 feet wide at ground level, what is the distance from the lowest point of the U-Dip to one of the points where the U-Dip meets ground level?
If a parabola modeling a water stream from a fireboat's cannon reaches a height of 50 feet above the barrel and the warehouse roof is 120 feet away, the value that indicates vertical shift in vertex form of equation is ______.
If a parabola modeling a water stream from a fireboat's cannon reaches a height of 50 feet above the barrel and the warehouse roof is 120 feet away, the value that indicates vertical shift in vertex form of equation is ______.
In the general form of a quadratic equation modeling the jackrabbit's jump, $y = a(x-h)^2 + k$, what do the variables h and k represent?
In the general form of a quadratic equation modeling the jackrabbit's jump, $y = a(x-h)^2 + k$, what do the variables h and k represent?
When modeling the path of a soccer ball with a parabola, the 'a' value in the quadratic equation will be positive if the parabola opens downwards.
When modeling the path of a soccer ball with a parabola, the 'a' value in the quadratic equation will be positive if the parabola opens downwards.
When modeling a real-world scenario with a parabola, such as the U-Dip, what does the domain of the parabolic function represent in the context of the problem?
When modeling a real-world scenario with a parabola, such as the U-Dip, what does the domain of the parabolic function represent in the context of the problem?
Match the scenario with the most appropriate point on the parabola to use for determining key parameters:
Match the scenario with the most appropriate point on the parabola to use for determining key parameters:
Flashcards
Asymptote (Function)
Asymptote (Function)
A line that a graph approaches but never touches as x approaches infinity or negative infinity.
Summary Statement (Function)
Summary Statement (Function)
A statement about a function supported by evidence from tables, equations, graphs, etc.
Turning Point (Parabola)
Turning Point (Parabola)
The highest or lowest point on a parabola, also known as the vertex.
Roots/Zeros (of a Graph)
Roots/Zeros (of a Graph)
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Y-Intercept
Y-Intercept
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Axis of Symmetry
Axis of Symmetry
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Reflection (Transformation)
Reflection (Transformation)
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Translation (Transformation)
Translation (Transformation)
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x → y Table
x → y Table
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Undefined x-value
Undefined x-value
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Asymptote
Asymptote
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Division by Zero
Division by Zero
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Graphing Points
Graphing Points
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Input Values Near h
Input Values Near h
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Sketching the Curve
Sketching the Curve
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Horizontal/Vertical Asymptotes
Horizontal/Vertical Asymptotes
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Identifying Quadratic Relationships
Identifying Quadratic Relationships
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Vertex of a Parabola
Vertex of a Parabola
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Roots/Zeros of a Quadratic Graph
Roots/Zeros of a Quadratic Graph
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Maximum vs. Minimum Vertex
Maximum vs. Minimum Vertex
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Axis of Symmetry from Vertex
Axis of Symmetry from Vertex
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X-Intercepts and Axis of Symmetry
X-Intercepts and Axis of Symmetry
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X-Intercepts and Quadratic Solutions
X-Intercepts and Quadratic Solutions
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Vertex Form
Vertex Form
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Quadratic Parent Function
Quadratic Parent Function
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Effect of 'a' in Vertex Form
Effect of 'a' in Vertex Form
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Effect of 'h' in Vertex Form
Effect of 'h' in Vertex Form
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Effect of 'k' in Vertex Form
Effect of 'k' in Vertex Form
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a>1 means?
a>1 means?
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a<0 means?
a<0 means?
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Finding Vertex Y-Value
Finding Vertex Y-Value
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Standard Form of Quadratic
Standard Form of Quadratic
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Factored Form of Quadratic
Factored Form of Quadratic
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Graphing Form
Graphing Form
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What is the vertex?
What is the vertex?
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X-Intercept(s)
X-Intercept(s)
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Exact Form (Radical Form)
Exact Form (Radical Form)
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Parabola
Parabola
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Quadratic Equation
Quadratic Equation
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Vertex
Vertex
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Vertex Form - 'h' Value
Vertex Form - 'h' Value
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Vertex Form - 'k' Value
Vertex Form - 'k' Value
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Vertex Form - 'a' Value
Vertex Form - 'a' Value
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Domain
Domain
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Range
Range
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Study Notes
- Booklet 2 focuses on quadratic equations and transformations.
- The booklet aims at teaching how to change the equation of a parabola to fit nonlinear data.
- It covers stretching, compressing, reflecting, and shifting the graph of f(x) = x².
- Recognizing transformations in graphing form aids in rewriting equations for easier graphing.
- Quadratic functions can model real-world situations like a rabbit's jump or a soccer ball's path.
Booklet Objectives
- Describe a function completely.
- Convert between standard and vertex forms of quadratic equations.
- Graph a quadratic equation.
- Explain how the equation shifts the parabola.
Lesson 1: Function Investigation
- Aims to describe functions completely.
- Focuses on graphing and investigating a family of functions with the form f(x) = 1/(x-h).
- In this equation 'h' can be any number.
- Multiple representations of functions are considered: tables, graphs, rules, and situations.
- The task involves choosing a value for h, creating an x → y table, and graphing the function.
- It's important to consider if additional information is needed to accurately represent the graph.
Further Guidance for Graphing
- Create an x → y table using integer x-values, ranging from 5 less than 'h' to 5 more than 'h'.
- Identify any x-values for which there is no corresponding y-value.
- Plot the points on a graph.
- Add more values to the table, especially those close to 'h', to observe the function's behavior near 'h'.
- Sketch the curve once enough points ensure an understanding of the graph's shape.
- Each team member chooses a different 'h' value to create and graph their own function.
Function Investigation Questions
- Inquire about the created functions as thoroughly as possible.
Math Notes: Graphs with Asymptotes
- Asymptotes are lines that a graph approaches but doesn't touch and can be diagonal lines or even curves.
- An asymptote requires some ideas from calculus.
- In this course, asymptotes will primarily be horizontal or vertical lines.
- A function has a horizontal asymptote when the graph approaches a certain y-value as x-coordinates move farther from zero.
- A graph has a vertical asymptote when y-coordinates get farther from zero as x-coordinates approach a certain value.
Investigating a Function, Part 2: Summary Statements
- A summary statement is a statement about a function along with thorough justification.
- Justifications should include multiple representations: x → y table, equation, graph, and situation if applicable.
- An example: y = x² has a range of all real numbers greater than or equal to zero (y ≥ 0).
- First justification: the graph's lowest point is on the x-axis.
- Second justification: the table shows no negative y-values.
- Third justification: squaring any number results in a positive answer.
Lesson 2: Parabola Transformations in Standard Form
- Aims at exploring the PHET quadratic simulator for standard form.
- Focuses on the special points of quadratic functions:
- Turning points (maxima, minima).
- Roots/zeros of the graph (x-intercepts).
- y-intercepts.
- Axis of symmetry.
- Predicts how the transformation and changing coefficients of a quadratic function will affect the graph.
- Identifies geometric transformations: reflection, translation, and dilation.
Exploring the PHET Simulation:
- Identify main discoveries and questions.
- Manipulate the coefficients of the equation and note their graphic effect.
- Determine when graphs are U-shaped vs. ∩-shaped.
- Use the simulation to describe a vertex, axis of symmetry and roots/zeros.
- Show how graph intersects with the x-axis
Lesson 3: Parabola Transformations in Vertex Form
- Explores using the PHET quadratic simulator for Vertex Form, which will
- Determine the location of the vertex from the equation and from a graph.
- Describe how the 'a' value transforms the equation.
- Quadratic Parent Function is y = x².
- Vertex form of a Quadratic is y = a(x − h)² + k
Using PhET:
- First, Open the Graphing Quadratics Phet simulator, then go to the 3rd screen to transform to vertex form.
- Determine information that the vertex form of a quadratic expression reveals and how. Complete tables describing the effects of the values a, h, and k on the parent function.
Predicting with Equations:
- Using quadratic functions, without graphing, predict if graph opens up or down using coordinates of vertex
Math Notes: Finding the Vertex
- Use formula x = -b/2a to find the x-coordinate of the vertex.
- Plug x-value back into the standard form equation to find y-value, and write as a point (x,y).
- Based on the parent function, make the parabola dilate, translate, and reflect.
Math Notes: Forms of Quadratics
- Lists three main forms of a quadratic equation: standard form, factored form, and graphing/vertex form.
- Assume a ≠0, meaning of a, b, and c are different for each row from below.
- Standard Form has y-intercept (0, c)
- Factored Form has x-intercepts are (-b, 0) and (-c, 0).
- Graphing/Vertex Form has Vertex form (h, k).
- Solutions to a quadratic equation can be written in exact form.
Lesson 4: Mathematical Modeling with Parabolas
- Transform parabolas around their axes and learn to write quadratic equations in graphing and standard form.
- Apply new skills by using parabolas and their equations to model situations.
Jumping Jackrabbits:
- The diagram at right shows a jackrabbit jumping over a three-foot-high fence, starting it's jump four feet away from a fence.
- The task involves sketching the situation and writing an equation that models the jackrabbit's path.
- Discussion points include:
- How to make a graph fit this situation.
- What information is needed to find an equation.
- How to ensure the equation fits the situation.
Modeling With Parabolas:
- Choose an equation (and values for domain and rage) to model the path of sketch.
- Choose where to place the x- and y-axes to make the problem easier.
- Label as many points as you can on your sketch.
- Explore using 2-65 Student eTool(Link on Canvas).
- Understand how to make a graph reflect a situation and points
Modeling Scenarios:
- Model the path of a skateboard after it has done the cement structure embedded into the found.
- Model the math of path of water from fire boat to fire in order to extinguish the fire, calculate maximums and minimum points depending on the path and travel of the vessel.
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