quadratics
TRANSCRIPT
1Algebra
QUADRATIC FUNCTIONS
• Quadratic Expressions, Rectangles and Squares• Absolute Value, Square Roots and Quadratic Equations• The Graph Translation Theorem• Graphing • Completing the Square• Fitting a Quadratic Model to Data• The Quadratic Formula• Analyzing Solutions to Quadratic Equations• Solving Quadratic Equations and Inequalities
2Algebra
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Quadratic – quadratus (Latin) , ‘to make square’
Standard form of a quadratic:
3Algebra
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Quadratic expressions from Rectangles and Squares
Suppose a rectangular swimming pool 50 m by 20 m is to be built with a walkway around it. If the walkway is w meters wide, write the total area of the pool and walkway in standard form.
Write the area of the square with sides of length in standard form
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Binomial Square Theorem
For all real numbers x and y,
Note: When discussing this, ask students whether any real-number values of the variable give a negative value to the expression. [ The square of any real number is nonnegative].
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Challenge
Have students give quadratic expressions for the areas described below.
1. The largest possible circle inside a square whose side is x.
2. The largest possible square inside a circle whose radius is x.
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Absolute Value, Square Roots and Quadratic Equations
In Geometry•
In Algebra
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Activity
1. Evaluate each of the following.
2. Find a value of x that is a solution to .
3. Find a value of x that is not a solution to .
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Absolute Value – Square Root Theorem
For all real numbers x,
Example 1
Solve
Example 2
A square and a circle have the same area. The square has side 10. What is the radius of the circle?
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ChallengeThe Existence of Irrational Numbers
Prove that cannot be written as a simple fraction.
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Graphs and Translations
Consider the graphs of and What transformation maps the graph of the first function
onto the graph of the second?
Graph – Translation Theorem
In a relation described by a sentence in x and y, the following two processes yield the same graph:
1. replacing by and by
2. applying the translation to the graph of the original relation.
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Example 1
Find an equation for the image of the graph of under the translation .
Corollary
The image of the parabola under the translation is the parabola with the equation
Example 2
a. Sketch the graph of
b. Give the coordinates of the vertex of the parabola
c. Tell whether the parabola opens up or down
d. Give the equation for the axis of symmetry.
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Graphing Suppose
a. Find when
b. Explain what each pair tells you about the height of the ball.
c. Graph the pairs over the domain of the function.
Note: Two natural questions about the thrown ball are related to questions about this parabola.
1. How high does the ball get? The largest possible value of h.
2. When does the ball hit the ground?
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Newton’s Formula
• is a constant measuring the acceleration due to gravity• is the initial upward velocity• is the initial height • the equation represents the height of the ball off the ground at time
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Completing the Square
Completing the square geometrically and algebraically
Theorem
To complete the square on .
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Practice
• Going back to , find the maximum height of the ball.• Rewrite the equation in vertex form. Locate the vertex of
the parabola.• Suppose
a. What is the domain of ?
b. What is the vertex of the graph?
c. What is the range of ?
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Fitting a Model to Data
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Practice
The number of handshakes needed for everyone in a group of people, to shake the hands of every other person is a quadratic function of Find three points of the function relating and Use these points to find a formula for this function.
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The Angry Blue Bird Problem
What if Blue Bird’s flight path is described by the function
Where is Blue Bird when she’s 8 feet high?
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The Quadratic Formula
If
Challenge
How was it derived?
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Practice
Solve
The 3-4-5 right triangle has sides which are consecutive integers. Are there any other right triangles with this property?
Challenge: Find a number such that 1 less than the number divided by the reciprocal of the number is equal to 1.
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How Many Real Solutions Does a Quadratic Equation Have?
Discriminant Theorem
Suppose are real numbers with
Then the equation has
i. two real solutions if
ii. one real solution if
iii. two complex conjugate solutions if .
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Practice
Determine the nature of the roots of the following equations. Then solve.
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Solving Quadratic Equations
Extracting Square Roots Factoring Completing the Square Quadratic Formula
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FactoringSteps:• Transform the quadratic equation into standard form if necessary.• Factor the quadratic expression.• Apply the zero product property by setting each factor of the quadratic
expression equal to 0.
Zero Product Property– If the product of two real numbers is zero, then either of the two
is equal to zero or both numbers are equal to zero.
• Solve each resulting equation.• Check the values of the variable obtained by substituting each in the original
equation.
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Practice
Solve the following equations.
1.
2.
3.
4.
5.
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HOW TO SOLVE?
1. find the "=0" points
2. in between the "=0" points, are intervals that are either
greater than zero (>0), or
less than zero (<0)
3. then pick a test value to find out which it is
(>0 or <0)
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Here is the plot of
The equation equals zero at -2 and 3
The inequality "<0" is true
between -2 and 3.
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Practice
1. Find the solution set of
2. Graph
3. A stuntman will jump off a 20 m building. A high-speed camera is ready to film him between 15 m and 10 m above the ground. When should the camera film him?