Precalculus Name:____________________________________Lesson- Graphing Quadratic Functions
Date:_____________________________________
Objective: To graph a quadratic function using the Roots Axis of symmetry Vertex
Do Now: Graph the following function.
Roots:
Axis of symmetry:
Vertex:
__________________________________________________________________________________________
Graph each of the following on the same set of axes by finding the intercepts, axis of symmetry and vertex.
Based on the graphs above, is there a shortcut for determining if the parabola opens up or down?
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x
y
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Precalculus Name:____________________________________Lesson- Completing the square and Quadratic Formula
Date:_____________________________________
Objective: To review finding the roots of a quadratic function by completing the square and quadratic formula.
Do Now: Find the roots of the quadratic function by using the zero-product rule.
Completing the square:
Examples:
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Discriminant:
Quadratic Formula:
Determine the nature of the roots of each of the following. In each case, determine which method of finding the roots of quadratics would be the best to use. Then find the roots of all the odd numbered problems by completing the square and the even numbered problems by using the quadratic formula. All answers should be in simplest form.
Precalculus Name:____________________________________
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HW: Completing the square
Showing all work, solve the following quadratic equations by completing the square:
(1) x2 – 2x – 24 = 0 (2) x2 – 2x – 2 = 0
(3) x2 – 8x + 17 = 0 (4)
(5) 4x2 – 16x + 20 = 0 (6) 4x2 – 4x – 5 = 0
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Showing all work, solve the following quadratic equations by completing the square:
(7) x2 – 6x + 25 = 0 (8) x2 – 3x – 18 = 0
(9) x2 – 2x + 10 = 0 (10)
(11) 2x2 + 15x – 8 = 0 (12) x2 – 10x + 23 = 0
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Precalculus Name:____________________________________Lesson- Standard form of a quadratic function
Date:_____________________________________
Objective: To write and graph a quadratic function in standard form
Do Now: Find the roots of the quadratic function by using the quadratic formula.
Standard form of a Quadratic Function:
where (h,k) is the vertex and a determines whether that vertex is a maximum or minimum.
Writing a quadratic in standard form:Example:
__________________________________________________________________________________________
Write each of the following in standard form and determine the vertex and whether that vertex is a maximum or minimum.
Precalculus Name:____________________________________
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HW- Standard form of a Quadratic Date:_____________________________________
Complete the square to change each quadratic function to the form: and state the vertex for each parabola. SHOW ALL WORK!
(Check your results using a graphing calculator.)
(1) f(x) = 3x2 + 12x – 2 (2) f(x) = 2x2 + 12x – 3
(3) f(x) = -2x2 + 6x + 7 (4)
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(5) f(x) = 4x2 – 20x + 9 (6) f(x) = 2x2 – 8x + 4
(7) f(x) = -x2 + 3x – 4 (8) f(x) = 3x2 + 27x + 5
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Precalculus Name:____________________________________Lesson- Synthetic Division and the Remainder Theorem
Date:_____________________________________
Objective: To use the synthetic division process as well as the remainder theorem to: Divide polynomials more quickly than the long division process Evaluate a function given the independent variable
Do Now: Find the roots of the following quadratic:
Synthetic Division List out the coefficients of the dividend in descending order (put a 0 in for any missing powers) Write the divisor in form and bring down the first coefficient of the dividend Multiply r by the 1st term and add to the second term Multiply r by the result of the previous step and add to the next term, repeat if necessary The last result is a remainder, treat it the same way you would in long division
Remainder Theorem is the remainder (coefficient) after the synthetic division process.
Examples:
1. Divide using synthetic division. 2.
3. Find if using the remainder theorem.
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Precalculus Name:____________________________________Lesson: Fundamental Theorem of algebra &
The Rational Root Theorem & The Date:_____________________________________Imaginary Root Theorem
Objective: to determine roots of polynomial equations by applying the Fundamental Theorem of Algebra to be able to use the rational root theorem to find the factors and roots of any polynomial to be able to use the imaginary root theorem to find complex roots of any polynomial
Fundamental Theorem of Algebra: In mathematics, the fundamental theorem of algebra states that every complex polynomial p(z) in one variable and of degree n ≥ 1 has some complex root.
Complex Numbers:
Examples:
Write the polynomial equation using the given roots:(1) 9 and 7 (2) -6 and 4 (3) -3 and -5
Write P(x) as a product of first-degree factors using the given zero:
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(1) P(x) = x3 + 9x2 + 24x + 16; -1 is a zero
(2) P(x) = 2x3 + 7x2 – 19x – 60; 3 is a zero
(3) P(x) = x3 – 4x2 – 3x + 18; 3 is a double zero
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Rational Roots Theorem : Possible roots , where p represents factors of the constant term and q
represents factors of the leading coefficient.
Imaginary Roots Theorem: Imaginary roots occur in complex conjugate pairs.
Process:1. Use Rational Roots Theorem to find potential rational roots.
2. Use synthetic division, or long division, to find an actual root.
3. Repeat step 2 until the polynomial is of degree 2.
4. Factor the remaining quadratic polynomial (it is possible to get imaginary answers).
5. List all factors for “completely factored form”.
More examples:1. 2.
3. 4.
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5. 6.
7. 8.
9.
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Write the polynomial equation using the given roots:
(10) 4, 6i, and -6i (11) 2, -5i, and 5i
Write P(x) as a product of first-degree factors using the given zero:
(12) P(x) = x3 – 16x2 + 86x – 156; 6 is a zero
(13) P(x) = x4 – 6x3 + 5x2 + 32x + 20; -1 is a double root.
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Precalculus Name:__________________________________Review for test- Polynomial Functions
Date:___________________________________
SHOW ALL WORK!(1) Write the polynomial equation with the given roots:
(a) 8 and -9 (b) 3, 4i, and -4
(c) i, -i, 5i, and -5i (d) 1, 0, and 2i
(2) WRITE P(X) AS A PRODUCT OF FIRST-DEGREE FACTORS USING THE GIVEN ZERO:(A) P(X) = 2X3 – X2 – 26X + 40; 2 IS A ZERO
(B) P(X) = 3X3 – 8X2 + 7X – 2; 1 IS A DOUBLE ZERO
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(C) P(X) = X3 – 9X2 + X + 111; -3 IS A ZERO
(3) SOLVE THE FOLLOWING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE:
(A) 2X2 – 5X – 12 = 0 (B) (C) 4X2 – 16X + 20 = 0
(4) COMPLETE THE SQUARE TO CHANGE THE FOLLOWING QUADRATIC FUNCTIONS TO THE FORM: and state the vertex for each parabola:
(A) (B) (C)
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(5) ALGEBRAICALLY SHOW IF X + 1 IS A FACTOR OF X5 + 1, AND EXPLAIN WHY OR WHY NOT.
(6) ALGEBRAICALLY SHOW IF X – 2 IS A FACTOR OF 2X4 – 3X, AND EXPLAIN WHY OR WHY NOT.
(7) FIND ALL OF THE FACTORS OF X3 – 6X2 + 25X – 150 IF ONE OF ITS FACTORS IS X – 6.
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(8) FIND ALL OF THE FACTORS OF X3 – 2X2 – 23X + 60 IF ONE OF ITS FACTORS IS X + 5.
(9) FIND ALL OF THE FACTORS OF X3 – 7X + 6 IF ONE OF ITS FACTORS IS X – 2.
(10) FIND ALL OF THE FACTORS OF X3 – 11X – 10 IF ONE OF ITS FACTORS IS X + 1.
(11) FIND ALL OF THE FACTORS OF
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