Quadratic Relations and Transformations

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Questions and Answers

What is the equation of the quadratic function in standard form with zeros of -3 and 1, passing through the point (-1, -7)?

  • y = -2x² - 4x - 3
  • y = -2x² + 4x - 3
  • y = -2x² - 4x + 3
  • y = -2x² + 4x + 3 (correct)

What is the equation of the quadratic function in vertex form with a vertex of (-2, 3) and passing through the point (-1, 6)?

  • y = 3(x - 2)² + 3
  • y = 3(x + 2)² + 3 (correct)
  • y = 3(x + 2)² - 3
  • y = 3(x - 2)² - 3

What is the equation of the quadratic function in factored form which has zeros of -2 and 4 and a y-intercept of -3?

  • y = -½(x + 2)(x - 4)
  • y = ½(x + 2)(x - 4)
  • y = ¾(x + 2)(x - 4)
  • y = -¾(x + 2)(x - 4) (correct)

What is the equation of the quadratic function in vertex form, with a maximum at (4, -2) and congruent to y = 2x²?

<p>y = -2(x - 4)² - 2 (B)</p> Signup and view all the answers

What is the equation of the quadratic function in standard form with a vertex of (-4, 1) and a y-intercept of -5?

<p>y = 1/8x² - 1/2x - 5 (A)</p> Signup and view all the answers

What is the step pattern of a parabola with the equation y = 2(x - 3)^2 + 1?

<p>Over 1, Up 2 (C)</p> Signup and view all the answers

What is the optimal value of the quadratic relation y = -x^2 + 4x + 5?

<p>9 (B)</p> Signup and view all the answers

Which of the following is NOT a characteristic of the axis of symmetry of a parabola?

<p>It determines the direction of opening of the parabola. (D)</p> Signup and view all the answers

What is the y-intercept of the quadratic relation y = 2x^2 - 3x + 1?

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

If the vertex of a parabola is located at the point (2, -3), what is the equation of the axis of symmetry?

<p>x = 2 (D)</p> Signup and view all the answers

What is the domain of the quadratic relation y = x^2 - 4x + 3?

<p>All real numbers (A)</p> Signup and view all the answers

What are the zeros of the quadratic relation y = x^2 - 4?

<p>x = 2, x = -2 (B)</p> Signup and view all the answers

If a parabola opens downwards, what can you conclude about the coefficient of the squared term in its equation?

<p>The coefficient is negative. (C)</p> Signup and view all the answers

What is the quadratic equation that models the number of imported cars sold in Newfoundland between 2003 and 2007?

<p>𝑦 = 76.85714𝑥 2 − 308177.94285714𝑥 + 308932906.6 (B)</p> Signup and view all the answers

Using the quadratic equation, how many imported cars would you expect to be sold in 2008?

<p>4516 (A)</p> Signup and view all the answers

What does the model predict for the number of imported cars sold in 2006?

<p>3862 (A)</p> Signup and view all the answers

In the quadratic inequality 𝑥 2 − 4 ≤ 0, what is the interval where the graph of the quadratic relation is on or below the x-axis?

<p>-2 ≤ x ≤ 2 (D)</p> Signup and view all the answers

What is the factored form of the quadratic inequality 𝑥 2 − 4 ≤ 0?

<p>(𝑥 + 2)(𝑥 − 2) ≤ 0 (A)</p> Signup and view all the answers

Using an interval chart to solve the quadratic inequality 𝑥 2 − 4 ≤ 0, where does the product of the factors (𝑥 + 2) and (𝑥 − 2) change signs?

<p>x = -2 and x = 2 (D)</p> Signup and view all the answers

In the context of solving quadratic inequalities, what does the term 'interval' refer to?

<p>A set of numbers between two specified numbers (C)</p> Signup and view all the answers

What is the main difference between solving a quadratic equation and a quadratic inequality?

<p>A quadratic equation results in a specific value for the variable, while a quadratic inequality results in an interval of possible values. (B)</p> Signup and view all the answers

A campground charges $20.00 to camp for one night and averages 56 people each night. If they decrease the price by $1.00, the number of campers increases by 7. What is the price that will maximize nightly revenue?

<p>$14.00 (D)</p> Signup and view all the answers

A toy rocket is launched with an initial velocity of 180 m/s and its height is modeled by h = -5t² + 180t, where t is the time in seconds. How long will the rocket stay above a height of 1000 meters?

<p>25 seconds (C)</p> Signup and view all the answers

A farmer has $5200 to spend on fencing for a pen along a river. The company charges $6.50 per meter for fencing. The farmer can choose between a rectangular pen with the river as one side or a right triangular pen with the hypotenuse along the river. Which shape maximizes the area of the pen?

<p>Both shapes have the same maximum area. (B)</p> Signup and view all the answers

The cost per book (C) when a school orders yearbooks is modeled by C = 0.00005n² - 0.095n + 66.125, where n is the number of books ordered. What is the least cost per book, and how many yearbooks should be ordered to achieve this cost?

<p>The least cost is $21 per book when 950 yearbooks are ordered. (A)</p> Signup and view all the answers

A ball is thrown into the air, and its height (h) in meters after t seconds is modeled by h = -5t² + 20t + 1. After how many seconds, rounded to two decimal places, does the ball hit the ground?

<p>4.05 seconds (D)</p> Signup and view all the answers

An object is thrown upward with an initial velocity of v m/s from an initial height of c meters. The height (h) after t seconds is modeled by h = -5t² + vt + c. What is the initial velocity of the object if it reaches its maximum height after 4 seconds?

<p>40 m/s (D)</p> Signup and view all the answers

The distance (d) a car travels while skidding is modeled by d = 250 + 50t - 4t², where t is the time in seconds taken to stop. How long does it take the car to stop?

<p>6.25 seconds (D)</p> Signup and view all the answers

The city bus company carries an average of 3500 passengers daily, with each passenger paying $2.25. If the company increases the price by $0.25, the number of passengers decreases by 100. What is the fare that will maximize the company's revenue?

<p>$2.75 (A)</p> Signup and view all the answers

What is the maximum daily revenue from coffee sales at The Next Cup coffee shop when the price per mug is optimized?

<p>$680 (D)</p> Signup and view all the answers

What was the height of the football at the moment it was kicked?

<p>1.1 m (A)</p> Signup and view all the answers

For the parabola defined by y = -2(x + 5)² - 4, what is the value of the minimum?

<p>-4 (C)</p> Signup and view all the answers

How many x-intercepts does the parabola y = -3x² have?

<p>0 (A)</p> Signup and view all the answers

What is the vertex of the parabola represented by y = 3x² + 18x + 21?

<p>(-3, 12) (B)</p> Signup and view all the answers

How long does it take for Baz Ketball's shot to reach its peak height?

<p>2 seconds (B)</p> Signup and view all the answers

In the function h = -5t² + 10t + 3, what is the distance above the floor when the ball reaches its peak?

<p>6 m (C)</p> Signup and view all the answers

What is the equation of the parabola that has a vertex at (2,4) and a y-intercept of -4?

<p>y = -1(x - 2)² + 4 (C)</p> Signup and view all the answers

What is the maximum revenue formula for a bus company in terms of price increases, given they lose customers?

<p>M = -12.5n^2 + 762.5n + 7875 (D)</p> Signup and view all the answers

What dimensions will maximize the area for a garden that uses 24 m of fencing on three sides?

<p>8 m by 4 m (A)</p> Signup and view all the answers

To maximize sales revenue, at what price should Tom sell his T-shirts if he currently sells them at $10?

<p>$9 (B)</p> Signup and view all the answers

At what price should Mila set her CDs, originally priced at $20 with a drop in sales for every increase?

<p>$19 (A)</p> Signup and view all the answers

How long will it take for the ball to reach a height of 35 m, given the height equation h = -4.9t^2 + 30t + 1.6?

<p>1.46 seconds (B)</p> Signup and view all the answers

When will a ball thrown into the air hit the ground if described by h = -4.9(t - 2)^2 + 20?

<p>4.02 seconds (C)</p> Signup and view all the answers

What is the height equation for a wrench tossed on the moon?

<p>h = –0.8t^2 + 10t + 1.4 (A)</p> Signup and view all the answers

After how many seconds will the wrench hit the ground according to its height equation?

<p>12.6 seconds (A)</p> Signup and view all the answers

Flashcards

Vertex

The highest or lowest point on the graph of a quadratic function, where the parabola changes direction.

Axis of Symmetry

The line that divides a parabola into two symmetrical halves.

Zeros

The values of x where the parabola intersects the x-axis.

Y-intercept

The point where the parabola intersects the y-axis.

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Optimal Value

The highest or lowest y-value that a parabola can have.

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Domain

All possible x-values for which a function is defined.

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Range

All possible y-values that a function can take on.

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Maximum/Minimum Value

The value of a function at its vertex.

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Standard Form of a Quadratic Function

A quadratic function is an equation of the form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0. It represents a parabola, which is a symmetrical U-shaped curve. The standard form allows us to readily identify the y-intercept and the coefficient 'a' which determines the parabola's direction (up or down).

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Vertex Form of a Quadratic Function (y = a(x - h)² + k)

The vertex form of a quadratic function is given by y = a(x - h)² + k, where (h, k) represents the vertex (the minimum or maximum point) of the parabola and 'a' controls the direction and width of the parabola. This form provides direct information about the vertex, which is crucial for sketching the graph.

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Factored Form of a Quadratic Function (y = a(x - r)(x - s))

The factored form of a quadratic function is expressed as y = a(x - r)(x - s), where 'r' and 's' are the x-intercepts (or roots) of the parabola. This form is beneficial for finding the x-intercepts and determining the direction of the parabola based on 'a'.

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Y-Intercept of a Quadratic Function

The y-intercept of a quadratic function is the point where the parabola crosses the y-axis. To find the y-intercept, set x = 0 in the equation and solve for y.

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Quadratic Equation

A mathematical expression that includes a variable raised to the power of 2, along with constants and other variables. It is used to represent various real-world phenomena like projectile motion, parabolic shapes, and optimization problems.

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Quadratic Inequality

A mathematical statement that involves a quadratic expression and an inequality symbol (<, >, ≤, ≥). It's used to determine the range of values of the variable that make the statement true.

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Graphical solution of Quadratic Inequalities

A method for solving quadratic inequalities graphically by plotting the corresponding quadratic function (equation) and identifying the regions where the graph satisfies the inequality (above or below the x-axis).

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Algebraic solution of Quadratic Inequalities

A method for solving quadratic inequalities by factoring the quadratic expression, identifying the critical points, and testing intervals to determine where the expression is positive or negative. This method helps determine the solution set for the inequality.

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What is the vertex of a parabola?

The highest or lowest point on the graph of a quadratic function, where the parabola changes direction.

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What is the axis of symmetry?

The line that divides a parabola into two symmetrical halves.

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What are the zeros of a parabola?

The values of x where the parabola intersects the x-axis.

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What is the y-intercept of a parabola?

The point where the parabola intersects the y-axis.

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What is the optimal value of a parabola?

The highest or lowest y-value that a parabola can have.

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What is the domain of a parabola?

All possible x-values for which a function is defined.

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What is the range of a parabola?

All possible y-values that a function can take on.

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What is the maximum/minimum value of a parabola?

The value of a function at its vertex.

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Time to Maximum Height

The time it takes for an object to reach its maximum height, when the object's height is modeled by a quadratic function.

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Initial Velocity

The initial velocity of an object is the speed at which it is launched or thrown, often represented by the coefficient of the linear term in a quadratic function modeling its height.

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Price Maximizing Revenue

The maximum revenue occurs when the price per night is set to a value that maximizes the total income from campsite rentals. This price can be found by using quadratic functions and finding the vertex.

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Time to Stop

The time it takes for a car to stop completely, when the car's distance traveled during skidding is modeled by a quadratic function.

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Time Above a Height

The time an object stays above a certain height, when the object's height is modeled by a quadratic function, is determined by finding the intervals where the height function exceeds that specific height.

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Optimal Number of Yearbooks

The optimal number of yearbooks to order to minimize the cost per book, when the cost function is a quadratic function, corresponds to the x-coordinate of the vertex of the parabola.

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Time to Hit the Ground

Finding the time an object hits the ground when its height is modeled by a quadratic function involves finding the x-intercepts of the parabola, where the height is zero.

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Maximizing Area of a Pen

The process of determining the optimal shape of a pen (e.g., rectangle vs. triangle) based on maximizing the area. This can be achieved by comparing the calculated areas of different shapes given fixed constraints.

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Study Notes

Quadratic Relations

  • Quadratic relations are relationships between variables that can be represented by a second-degree polynomial.
  • The general form of a quadratic relation is y = ax² + bx + c, where 'a', 'b', and 'c' are constants.
  • A quadratic relation has a graph that is a parabola.
  • A parabola can open upwards or downwards depending on the value of 'a'.
  • If 'a' is positive, the parabola opens upward.
  • If 'a' is negative, the parabola opens downward.
  • The vertex is the highest or lowest point on the parabola.
  • The axis of symmetry is a vertical line that divides the parabola into two symmetrical halves.
  • The zeros of a quadratic relation are the x-intercepts (the points where the graph crosses the x-axis).
  • The y-intercept is the point where the graph crosses the y-axis.
  • The domain of a quadratic relation is the set of all possible x-values.
  • The range of a quadratic relation is the set of all possible y-values.

Transformations of Quadratics

  • Transformations of quadratic functions involve shifting, stretching, compressing, or reflecting the graph of the basic quadratic function (y = x²).
  • The vertex form of a quadratic function is y = a(x - h)² + k, where (h, k) represents the coordinates of the vertex.
  • The value of 'a' affects the vertical stretch or compression of the parabola.
  • A positive value of 'a' indicates an upward opening parabola.
  • A negative value of 'a' indicates a downward opening parabola.
  • The value of 'h' affects the horizontal translation of the parabola.
  • A positive value of 'h' shifts the graph to the right.
  • A negative value of 'h' shifts the graph to the left.
  • The value of 'k' affects the vertical translation of the parabola.
  • A positive value of 'k' shifts the graph upward.
  • A negative value of 'k' shifts the graph downward.

Applications of Quadratics

  • Quadratic functions can model various real-world situations, including projectile motion, optimization problems, and business scenarios.
  • A range of word problems can be solved using quadratic relations such as profit maximization, projectile motion, or area maximisation

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