Quadratic Functions and Transformations
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Refer to the graph shown. Write down the coordinates of the vertex.

(4, 1)

Refer to the graph shown. Write the equation in the form y = 3(x − h)² + k

y = 3(x - 4)² + 1

Refer to the graph shown. Write down the equation of the axis of symmetry.

x = 4

Refer to the graph shown. Write down the domain and range.

<p>D: {x: x ∈ R}, R: {y: y ≥ 1}</p> Signup and view all the answers

Write down the coordinates of the vertex.

<p>(0, -1)</p> Signup and view all the answers

Sketch the graph of the parent quadratic, y = _x_², and the graph of y = g(x) on the same axes. Then write down the coordinates of the vertex and the equation of the axis of symmetry for the graph of g. g(x) = (x + 3)²

<p>Vertex: (-3, 0), Axis of symmetry: x = -3</p> Signup and view all the answers

Sketch the graph of the parent quadratic, y = _x_², and the graph of y = g(x) on the same axes. Then write down the coordinates of the vertex and the equation of the axis of symmetry for the graph of g. g(x) = −_x_² + 4

<p>Vertex: (0, 4), Axis of symmetry: x = 0</p> Signup and view all the answers

Sketch the graph of the parent quadratic, y = _x_², and the graph of y = g(x) on the same axes. Then write down the coordinates of the vertex and the equation of the axis of symmetry for the graph of g. g(x) = 2(x − 4)² − 3

<p>Vertex: (4, -3), Axis of symmetry: x = 4</p> Signup and view all the answers

Describe the transformations of the graph of f(x) = _x_² that lead to the graph of g. Then write an equation for g(x).

<p>g(x) = (x - 3)² + 4, The transformations are a horizontal translation 3 units to the right and a vertical translation 4 units upward.</p> Signup and view all the answers

What are the x-intercepts of the quadratic function f(x) = (x - 3)^2 - 2?

<p>3 and 1 (E)</p> Signup and view all the answers

Find the coordinates of the vertex of the quadratic function f(x) = -2x^2 + 4x - 8?

<p>(1, -6)</p> Signup and view all the answers

Match the following transformations with the corresponding effect on the graph of the quadratic function f(x) = x^2:

<p><code>f(x - 2)</code> = Translate the graph 2 units to the right <code>-f(x)</code> = Reflect the graph across the x-axis <code>f(x) + 3</code> = Translate the graph 3 units upward <code>2f(x)</code> = Vertically stretch the graph by a factor of 2</p> Signup and view all the answers

Find the equation of the axis of symmetry for the function f(x) = 3x^2 + 18x + 20?

<p>x = -3</p> Signup and view all the answers

What are the x-intercepts of the quadratic function f(x) = x^2-12x+36?

<p>x = 6</p> Signup and view all the answers

Find the coordinates of the point of intersection between the graphs of the functions f(x) = x^2 - 8x + 5 and g(x) = 3x^2 - 6x + 2?

<p>(1, -2)</p> Signup and view all the answers

The graphs of the functions f(x) = x^2 - 8x + 5 and g(x) = -2x^2 - 8x - 11 intersect at two points.

<p>False (B)</p> Signup and view all the answers

What is the equation of the axis of symmetry for the quadratic function f(x) = 2(x + 3)(x - 1)?

<p>x = 1</p> Signup and view all the answers

Find the coordinates of the vertex of the quadratic function f(x) = -3(x - 2)^2 + 5?

<p>(2, 5)</p> Signup and view all the answers

What are the x-intercepts of the quadratic function f(x) = 2x^2 + 6x + 3?

<p>x = -1/2, x = -3</p> Signup and view all the answers

What is the equation of the axis of symmetry of the function f(x) = 4(x + 3)(x - 1)?

<p>x = -1</p> Signup and view all the answers

Flashcards

Vertex of a Parabola

The turning point of a parabola, representing the minimum or maximum value of the function.

Axis of Symmetry

A vertical line that divides a parabola into two symmetrical halves.

Parabola Equation (Vertex Form)

𝑦 = 𝑎(𝑥−ℎ)^2 + 𝑘 where (h,k) is the vertex and a affects the shape.

Domain

The set of all possible input values (x-values) for a function.

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Range

The set of all possible output values (y-values) for a function.

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Quadratic Function

A function that can be written in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

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Parabola Intercept

The point(s) where the parabola crosses the x-axis (y = 0) or y-axis (x = 0).

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Maximum Height

The highest point a projectile reaches from a given function

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Time of Flight

The total time taken for a projectile to reach the ground

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Projectile Motion

Motion of an object thrown or projected into space

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Height of Projectile

The vertical distance a projectile travels.

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Initial Velocity

The speed and direction of a projectile at its launch.

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Initial Height

The vertical position of the projectile at its launch.

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Horizontal Distance

The distance traveled by a projectile along the horizontal plane.

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Time

Duration for a projectile journey

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Assembly Rate

The number of items produced per unit of time

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Parabola

The graph of a quadratic function, shaped like a symmetric 'U' or upside-down 'U'.

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Vertex

The highest or lowest point on a parabola, representing the minimum or maximum value of the function.

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Standard Form (Quadratic)

A quadratic function written as f(x) = a(x - h)² + k, where (h, k) represents the vertex.

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Vertex Form (Quadratic)

A form of a quadratic function that quickly reveals the vertex and axis of symmetry.

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Intercept Form (Quadratic Function)

A quadratic function written as f(x) = a(x - p)(x - q), where 'p' and 'q' represent the x-intercepts.

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X-Intercepts (Quadratic Function)

The points where the parabola crosses the x-axis, also known as the roots or zeros.

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Quadratic Equations

Equations formed by setting a quadratic function equal to zero, typically written as ax² + bx + c = 0.

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Points of Intersection

The points where two or more graphs of functions share a common point; the values at which the equations have the same solution.

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Solving Quadratic Equation

Finding the solutions to a quadratic equation, often by factoring or using the quadratic formula.

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Quadratic Formula

A formula used to solve quadratic equations: x = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are the coefficients in the equation.

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Factoring (Quadratic Equations)

A method to solve quadratic equations by breaking down the expression into two binomials that multiply to the original quadratic equation.

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Solution to Quadratic Equation

The value(s) that satisfy the quadratic equation, often representing the x-intercepts of the function.

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Real Solutions (Quadratic Equation)

Solutions to a quadratic equation that are real numbers, meaning they are represented on a number line.

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No Real Solutions (Intersecting Graphs)

When a quadratic function and another function, like a line, do not intersect.

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Applications of Intersections

Using the points of intersection to analyze real-world situations involving quadratic functions, such as analyzing projectile motion or economics.

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Supply and Demand

The relationship between the quantity of goods produced (supply) and the quantity consumers want (demand), often modeled using quadratic functions in economics.

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Understanding Quadratic Functions

Comprehending the properties, graphs, and applications of quadratic functions, enabling the solving of real-world problems involving these functions.

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Modeling Real-World Phenomena

Using quadratic functions to represent and analyze real-world situations, such as the trajectory of a projectile, the shape of a bridge, or the profit of a business.

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Problem-Solving Strategies

Utilizing various methods to approach and solve problems involving quadratic functions, including graphing, factoring, using the quadratic formula, and interpreting the results.

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Interpreting Results

Analyzing the solutions obtained from solving quadratic equations or finding the points of intersection to draw meaningful conclusions about the problem.

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Coordinate System

A system of two perpendicular lines (x-axis and y-axis) used to plot points and represent relationships between variables.

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Cartesian Plane

Another name for the coordinate system, named after René Descartes.

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Ordered Pair

A pair of numbers, (x, y), that represents the coordinates of a specific point on the coordinate plane.

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Graphing Techniques

Methods used to plot the graphs of functions, including plotting points, finding intercepts, and using the properties of the functions.

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Mathematical Modeling

Using mathematical concepts and equations to represent and solve problems in various fields, such as physics, engineering, and economics.

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Solving for X

Finding the value(s) of 'x' for a given equation, often using algebraic techniques like factoring or the quadratic formula.

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Solving for Y

Finding the value(s) of 'y' for a given equation, often by substituting a known value for 'x' or using other algebraic techniques.

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Algebraic Techniques

Methods used to manipulate equations and solve for unknown variables, such as factoring, expanding expressions, and using the quadratic formula.

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Study Notes

Quadratic Functions and Transformations

  • Various quadratic functions are presented, along with their corresponding graphs.
  • Key features, such as vertex coordinates, intercepts, axis of symmetry, domain, and range, are identified for each function.
  • Students need to label these key features on sketches of the functions.

Exercise 3J

  • Problems involve sketching parent quadratic functions (y = x²) and related functions (g(x)) on the same axes.
  • Coordinates of the vertex and the axis of symmetry need to be determined for each transformed quadratic.

Exercise 3L

  • Students use a graphing calculator (GDC) to plot quadratic functions.
  • Key features (x-intercepts, y-intercepts, vertex) need to be identified and labeled on the graph.
  • Domain and range of the functions are required.

Exercise 3M

  • Problems focus on determining the equation of symmetry, coordinates of the vertex, and y-intercept of various quadratic functions.

Exercise 3N

  • Students are required to express quadratic functions in the form f(x) = a(x - p)(x - q), where p > q.
  • Finding the x-intercepts and y-intercept coordinates of the function is also part of this exercise.

Exercise 3R

  • Finding the exact values and graphical solutions for quadratic equations.
  • Determining the points of intersection of specified quadratic and linear functions through graphical methods.

Exercise 3W

  • Solving quadratic inequalities graphically.
  • Identifying the values of constants that cause quadratic equations to have two distinct real roots or no real roots.

Exercise 3X

  • The length of the base of a triangle is related to its height and area. A function models the ball's height over time.

Exercise 30

  • Finding the expression for quadratic functions using given information from their graphs.

Exercise 3P

  • Various problems involving quadratic functions and their transformations.

Exercise 3S

  • A variety of problems related to quadratic functions require finding key features such as roots, vertex, and symmetry. A variety of quadratic problems.

Exam-Style Questions

  • Problem types related to quadratic functions and their transformations appear throughout the exercises.
  • Finding solutions and describing these types of transformations of the functions are frequent tasks.

Review

  • Problems involve identifying features of graphs of quadratic functions and calculating important components like the vertex, axis of symmetry, and intercepts, as well as describing transformations.

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Related Documents

Quadratic Functions Assignment 2023 PDF

Description

Test your knowledge on quadratic functions and their transformations. This quiz covers key features like vertex coordinates, intercepts, domain, and range. You'll sketch graphs, identify characteristics, and utilize graphing calculators to deepen your understanding of these essential concepts.

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