Make-Up Trigonometry Evaluation

Math 12V: Quadratic Functions Review

Here is a list of the major topics that we’ve covered in this chapter. For each topic I’ve included in brackets additional questions from the text. Some of these we did together in class and others would be good additional practice.

Sequences: You should be able to identify and describe the properties of arithmetic, quadratic, cubic, quartic and geometric sequences. You should be able to determine the equation of an arithmetic, quadratic or geometric sequence algebraically. You should also be able to determine the equation of any sequence using regression on a graphing calculator. (p4 #8, p9 #25, p10 #27, p12 #38, p13 #39, p112 #9)

Name		How do I recognize it?	General Term
Arithmetic		The sequence of first-level differences, , are all the same.	or
Power Sequences	Quadratic	The sequence of second-level differences, , are all the same
	Cubic	The sequence of third-level differences, , are all the same
	Quartic	The sequence of fourth-level differences, , are all the same
Geometric		There is a common ratio,

Example Problem #1: Determine the general term (t_n = ?) for each sequence below. Use the equation to find the 20^th term of the sequence.

a) b) c)

Forms of a Quadratic Function: A quadratic function can be written in general, standard or transformational form. You should be able to algebraically change from one from to any other form. You should also be able to find the vertex, axis of symmetry, y-intercepts and the domain and range of any quadratic function. (p10 #28, 29, 30, p13 #39, 40, p18 #19, 20)

General Form

Standard Form

Transformational Form

a : stretch

c : y-intercept

a : stretch

k : vertical translation

h : horizontal translation

(h, k) : vertex

x = h : axis of symmetry

Example Problem #2: Write each equation in transformational form. Find the vertex, axis of symmetry, y-intercept and domain and range for each.

a) b)

Finding the Formula of a Parabola: If you are given the vertex of a parabola and any other point on the curve, you can determine the formula for the parabola using the transformation for of the quadratic. The vertex is (h, k) and the other point is (x, y). When you plug the numbers into the equation, you can solve for a (the stretch).

Example Problem #3: What is the equation of a parabola with a vertex at (-3, 4) and a y-intercept of (0, -2)?

Finding Maximum and Minimum Values: You be able to solve a word problem to find the maximum or minimum value of a quadratic equation. (p22 #24-27; p34 #27, 28; p35 #29, 30, 31)

Example Problem #4: The perimeter of a parallelogram is 36 cm. The height of the parallelogram is 2 less than the length of one of its sides. Find the lengths of the sides of the parallelogram with maximum area.

Example Problem #5: What is the largest value that can be obtained by multiplying two real numbers whose sum is three?

Finding the Roots of a Quadratic Equation: The roots, or “zeros,” of a quadratic equation are the values of x that make the equation equal to zero. The real roots of a quadratic function are the same as the x-intercepts of the graph of the function. If a function has non-real roots, then the graph has no x-intercepts. We learned three different algebraic methods to find the roots of a quadratic: factoring, completing the square and the quadratic formula. You can also find the x-intercepts of a quadratic function using a graphing calculator. (p49 #30)

The Quadratic Formula

when

Example Problem #6: Solve each of the following equations. If the roots are non-real, express them as complex numbers.

a) b) c)

The Discriminant - What a Functions Roots Tell you about Its Graph - A quadratic function has two roots and you can find out what type of roots they are by using the discriminant = .

A Quadratic function with...	two real roots	a double root (two roots of the same value)	two non-real roots
has a discriminant...	positive discriminant	zero discriminant	negative discriminant
looks like this ...
and has this many x-intercepts	two x-intercepts	one x-intercept.	zero x-intercepts

Example Problem #7 : For what values of k, does the equation have two distinct real roots?

Using the Quadratic Formula to Solve Word Problems – You should be able to solve word problems involving quadratic relationships. You should be able to find the x-intercepts, y-intercept or the vertex of a quadratic from a written description of the relationship.

Example Problem #8: Jimmy throws his calculator out of the window of Mr. Lee’s classroom. The equation gives the height, in meters, of the calculator above the ground t seconds after it leaves his hand.

a) What is the maximum height of the calculator?

b) How long does it take for the calculator to hit the ground?

Example Problem #9: A 7m by 7m square swimming pool is surrounded by a path of uniform width. If the area of the path is 62m², find the width of the path.

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Solutions to Example Problems: If you spot any errors, please let me know so that I can correct them.

1a)

1b)

1c)

2a)

*** thanks to Wilson who fixed a mistake in this answer. ***

vertex: (4.5, -9.25)

axis of symmetry: x = 4.5

y-intercept: (0, 11)

domain:

range: or

2b)

vertex: (2.5, -4.5)

axis of symmetry: x = 2.5

y-intercept: (0, -17)

domain:

range: or

3) The vertex is (-3, 4) so h = -3 and k = 4. The y-intercept (0, -2) is a point on the parabola so x = 0 and y = -2. Now use the transformational form of a quadratic to solve for a.