Download Quadratic Functions: Review

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Algebra 2
Quadratic Functions: Review
1. Write f(x) formulas for quadratic functions that have the specified zeros.
(For most of the functions, it might be easiest to write a factored form first.)
Then, convert the formulas into standard (ax2 + bx + c) form.
a. with zeros 3 and –2
b. with zeros –4 and
2
3
c. with zeros –2.5 and 0
d. with 5 as the only zero
e. with no zeros
Hint: Can’t start from factored form for this one.
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Algebra 2
2. Find the (x, y) coordinates of the vertex of these functions, using the specified methods.
a. f(x) = x2 – 8x – 4, by putting the function into vertex form using completing the square.
b. f(x) = –2x2 + 12x – 10, by factoring to find the zeros, then using the zeros
c. f(x) = 2x2 – 7x + 13, using the formula that contains a and b
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2. (continued) Find the (x, y) coordinates of the vertex of these functions…
d. h(t) = –16t2 + 50t + 10, graphically on the calculator
e. f(x) = –3x2 – 12x + 8, by putting the function into vertex form
f. f(x) = –4(x + 3)2 + 5, using the easiest method available
g. f(x) = 2(x – 3)(x – 7), using the easiest method available
h. f(x) = 42x2 – 10x + 19, using your choice of method
Algebra 2
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3. Do all of the following for the function f(x) = –2x2 + 4x + 6.
a. Using factoring, find the zeros.
b. Using completing-the-square, find the vertex.
c. Find the y-intercept.
d. Draw the 4 points found so far, then
sketch the graph of f(x).
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4. Answer these questions for the function f(x) = –3(x + 2)2 + 9.
a. What are the coordinates of the vertex?
b. What is the equation for the axis of symmetry?
c. Find the y-intercept.
d. Using an algebraic (non-graphical) method, find the zeros.
e. Sketch a graph of f(x) using the
information found above.
Algebra 2
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Algebra 2
5. Solve these equations algebraically (not graphically). If a method is stated, you must use that
method. Show your solving steps.
a. 10 x 2  33x  7 . Solve by factoring.
b.  2 x 2  12 x  18 . Solve by completing the square.
c.  6x2x  1  0
d.  2x  12  5  1
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6. Find a quadratic function formula for each set of information given below.
a. The zeroes of the function are -3 and 6, and f(3) = 6
b. The minimum point of the parabola is (-2, 4) and f(0) = 12
Algebra 2
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Algebra 2
Answers to “Quadratic Functions: Review”
1. a. f(x) = (x – 3)(x + 2) = x2 – x – 6.
b. f(x) = (x + 4)(x –
2
3
) = x2 +
10
3
x–
8
3
or f(x) = (x + 4)(3x – 2) = x2 + 10x – 8.
c. f(x) = (x + 2.5)(x – 0) = x2 + 2.5x.
d. f(x) = (x – 5)2 = x2 – 10x + 25.
e. There are many correct answers. Pick one of these strategies:
o Write f(x) = ax2 + bx + c picking any numbers a, b, and c that make b2 – 4ac negative.
o Write f(x) = a(x – h)2 + k picking (h, k) to be any point above the x-axis, and a > 0.
o Use the fact that x2 is never negative to write a formula that can’t possibly equal 0,
such as f(x) = x2 + 1.
2. a. f(x) = (x – 4)2 – 20; vertex (4, –20).
b. Zeros are 1 and 5, whose average is 3; vertex (3, 8).
c. x = -
b
2a
=
7
4
; vertex (1.75, 6.875).
d. Vertex ≈ (1.56, 49.06).
e. f(x) = –3(x + 2)2 + 20; vertex (–2, 20).
f. It’s in vertex form, so vertex (–3, 5).
g. Easiest to average the zeros; vertex (5, –8).
h. Graphically on the calculator is probably
easiest; vertex (0.12, 18.40).
3. a. f(x) = (–2x + 6)(x + 1); zeros 3 and –1.
b. f(x) = –2(x – 1)2 + 8; vertex (1, 8).
c. f(0) = 6.
d. See graph at right.
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Algebra 2
4. a. (–2, 9)
b. x = –2
c. f(0) = –3.
d. x = -2 ± 3
e. Graph shown at the right, includes points
(–2, 9), (–3.73, 0), (–0.27, 0), and (0, –3).
5.
a.
2 x  75 x  1  0,
therefor e x 
7
1
,x
2
5
b. x  3  18  3  3 2 (second form is not required)
c. x = 0, x = ½
d. x  1  2 (Taking square roots is easiest method.)
6. a.
b.
1
f (x) = - (x + 3)(x - 6)
3
f (x) = 2(x + 2)2 + 4