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I. Quadratic Functions
A. A ___________________ function is a function of the form f(x)=ax^2+bx+c, where a, b and c are real numbers with a ≠ 0.
1. The graph is a U-shaped curve called a ____________________.
2. the parabola will “open up” if ____________ and it will “open down” if ___________.
B. f(x)=a(x-h)^2+k is the ________________ form of a quadratic function
1. The point (h, k) is the _______________ of the parabola.
2. Given the quadratic function f(x)=2(x-3)^2-2:
a. the vertex is __________
b. the parabola opens _________ (up or down)
c. the axis of symmetry is the line ____________.
d. the y-intercept is ________.
e. the x-intercepts are _______ and ______.
C. Rewrite the quadratic function f(x)=5x^2-30x+49 in standard form.
1. Factor the leading coefficient from the first two terms,
f(x) = 5(x2 – ____x) +49
2. To complete the square inside the parentheses we would add ______
3. If we add 9 inside the parentheses (which will be multiplied by the 5), we need to subtract ______ outside the parentheses.
f(x)=5(x^2-6x+9)+49-45
4. Factoring the trinomial and simplifying like terms gives
f(x) = 5(x – ____)2 + 4
5. The vertex of the parabola is _________.
D. Using the Vertex Formula
1. The first coordinate of the vertex, h, can be obtained directly from the form f(x)=ax^2+bx+c
a. we get h=-b/2a
2. Once we have h, we can find k by evaluating the function at h, f(h).
3. Use the vertex formula to find the vertex, (h, k), of the parabola defined by f(x)=2x^2-5x-3
a. h = ______
b. k = f(h) = _______
II. Applications of Quadratic Functions
A. An object is launched vertically in the air from a 10-meter-tall platform. The height (in meters) of the object t seconds after it was launched is modeled by the function h(t)=-16t^2+36t+10. How long will it take for the object to reach its maximum height? What is the maximum height?
1. If we graphed the function, we would get a parabola that opened down and therefore has its maximum value at the ____________.
a. The t-coordinate of the vertex is given by t=-b/2a = _________.
b. The object reaches its maximum height _______ seconds after launch.
2. The maximum height will be the value of the function at the vertex.
a. h(1.25) = ___________
b. The maximum height of the object is ___________ meters.
III. Finding Maximum/minimum values using a graphing calculator.
A. Go to the Graphing Calculator website: http://dtc.pima.edu/~dwilliamson/TI/indexti.html and view the topic “CALC Key – Max & Min”
B. The find the maximum value of the function f(x)=-4.9x^2+44.1x+1:
1. Enter the equation using the [Y=] key. (Careful to use the negative and not the minus sign in the first term.)
2. Find the correct viewing window to get a good look at the shape of the curve. [-2,10] x [0,110] works well with this function
3. Go into the CALC menu by pressing [2nd][CALC]. (CALC is the 2nd function of the TRACE key.)
4. Press [4] for maximum.
5. The calculator will ask for a left boundary. Move your cursor along the curve to the left of the vertex using the left arrow key. Then press [ENTER].
6. The calculator will ask for a right boundary. Move your cursor along the curve to the right of the vertex using the right arrow key. Then press [ENTER].
7. Press [ENTER] again to have the calculator guess (calculate).
8. The coordinates of the vertex are given at the bottom of the screen.
C. Find the maximum value of the function g(x)=-x^4-x^2+x+2 using a graphing calculator. The maximum value is _____________.
IV. Graphs of Polynomial Functions
A. The function f(x)=a_n x^n+a_(n-1) x^(n-1)+a_(n-2) x^(n-2)+ ∙∙∙+a_1 x+a_0 is a ___________________________ function of degree _____. The numbers a0, a1, a2,…,an are called the ______________________ of the function. The number an is called the ______________________ coefficient, and a0 is called the ___________________ coefficient.
B. Monomial functions of the form f(x) = axn are called ___________________ functions.
C. Determining the End Behavior of Polynomial Functions
1. The nature of the graph of a polynomial function for large values of x in the positive and negative direction is known as the ____________________________.
2. The end behavior of the graph depends on the ____________________ term.
3. Determine the sign of the leading ____________________.
a. if the sign is positive, the right-hand behavior “finishes ___________.”
b. if the sign is negative, the right-hand behavior “finishes ___________.”
4. Next, determine the ______________.
a. if the degree is __________, the graph has the same left-hand an right hand end behavior.
b. if the degree is __________, the graph has opposite left-hand an right hand end behavior.
5. For example, if we graph the function p(x)=-3x^4+5x^3-2x^2-4x+20:
a. the right-hand behavior finishes ____________.
b. the left-hand behavior finishes ____________.
D. Determining the Intercepts of a Polynomial Function
1. Every polynomial function, y = f(x), has a y-intercept that is found by evaluating ______.
2. To find the x-intercepts we find the real solutions to the equation f(x) = _____.
3. If f(c) = 0, then c is called a ___________ of the function f.
4. Find the intercepts of the polynomial function h(x)=x^3+3x^2-25x-75.
a. To find the y-intercept we evaluate h(0) = 03 +3(0)2 – 25(0) – 75 = ________
b. To find the x-intercept we write h(x) = 0
x^3+3x^2-25x-75=0
c. Factor by grouping
x2(____________) – 25(____________) = 0
d. Factor out the common factor (x + 3)
(x + 3)(x2 – 25) = 0
e. Factor using the difference of squares
(x + 3)(x + ____)(x – _____) = 0
f. Using the zero product property, x = ______, ______, or _______
g. These are the x-interepts
E. Determining Zeros and Multiplicities
1. If f is a polynomial function and c is a zero, then __________ is a factor.
2. If (x – c)k is a factor of f, then c is a zero of multiplicity ______.
3. If the multiplicity of a zero is ________ , then the graph crosses the x-axis at the zero.
4. If the multiplicity of a zero is ________ , then the graph touches the x-axis at the zero, but does not cross it.
F. Sketching the Graph
1. A polynomial of degree n, has at most _________ turning points.
2. A turning point in which the graph changes from decreasing to increasing is called a ___________________________.
3. A turning point in which the graph changes from increasing to decreasing is called a ___________________________.
4. Using a graphing calculator, graph the function f(x)=-2x^3-4x^2+2x+4.
a. Find the y-intercept. _________
b. Find the x-intercepts. ________, _________, and ___________
c. Are the multiplicities of these zeros even or odd? __________
d. Find the relative minimum value. ___________
e. Find the relative maximum value. __________
f. The right-hand end behavior finishes ____________.
g. The left-hand end behavior finishes ____________.
h. How many turning points are there?
V. Synthetic Division
A. The equation f(x) = d(x)q(x) + r(x) is used to check that long division was done properly. The original polynomial should equal the product of the divisor and the __________________ plus the _____________________. This process is known as the division _____________________.
B. We can use a shortcut to long division of polynomials called synthetic division if the divisor d(x) is in the form ______________.
1. If the divisor is x – 8, then c = _________.
2. If the divisor is x + 6, then c = _________.
C. Remainder Theorem: If a polynomial f(x) is divided by x – c, then the remainder is ________.
D. Factor Theorem: x – c is a factor of the polynomial f(x) if and only if f(c) = ______.
1. Use synthetic division, the Remainder Theorem and the Factor Theorem to determine if x + 1 is a factor of 2×3 – 5×2 + 7x + 1
-1 | 2 -5 7 -4
-2 7 -14
2 -7 14 -18 The remainder is _________, so is x + 1 a factor? _______
E. Given that -1 is a zero of f(x)= -x^3+5x^2-3x-9, find the remaining zeros and write f(x) in completely factored form.
1. Use synthetic division with c = _______
-1 | -1 5 -3 -9
1 -6 9
-1 6 -9 0
2. The quotient is ____________________
3. Using the division algorithm, f(x) = (x + 1)(-x2 + 6x – 9)
4. Factor a -1 from the trinomial to get (x + 1)(-1)(x2 – 6x + 9)
5. Finally, factoring the trinomial leaves us with (x + 1)(-1)( )2
6. The zeros of the function are -1 and _________.
VI. Zeros of Polynomial Functions
A. Rational Zeros Theorem
1. If a polynomial has integer coefficients, we are able to create a list of the potential ____________________ zeros.
2. If p/q is a rational zero of a polynomial with integer coefficients, then p must be a factor of the ______________________ coefficient, and q must be a factor of the ________________________ coefficient.
3. Use the Rational Zeros Theorem to determine the potential rational zeros of f(x)=3x^4-7x^2-x+8.
a. p must be a factor of _______.
b. so the possible values of p are ±1, ±2, ±4, or ±______
c. q must be a factor of _______.
d. so the possible values of q are ±____ or ±3.
e. so the potential rational zeros are every combination of p/q , or
±1, ±2, ±4, ±8, ±1/3, ±2/3, ±4/3, or ± _______
B. Number of Zeros Theorem
1. Every polynomial of degree n has _____ complex zeros provided each zero of multiplicity greater than 1 is counted accordingly.
2. Find all the complex zeros of f(x)=2x^4-5x^3-11x^2+20x+12.
a. There will be _____ zeros.
b. The possible rational zeros are: ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2
c. Use synthetic division and the remainder theorem we test each to see if it is a zero.
d. When we get to 2, we see that the remainder is zero and therefore (x-2) is a factor.
2 | 2 -5 -11 20 12
4 -2 -26 -12
2 -1 -13 -6 0
e. Since the bottom row gives us the quotient, according to the division algorithm
2x^4-5x^3-11x^2+20x+12=(x-2)(2x^3-x^2-13x-6)
f. We pick up with the list of potential rational zeros where we left off, x = 2. Checking 2 as a possible zero of the quotient, 2x^3-x^2-13x-6, we see it is not a zero. We continue through the list until we get to 3. When we get to 3, we see that the remainder is zero and therefore (x-3) is a factor.
3 | 2 -1 -13 -6
6 15 6
2 5 2 0
g. Since the bottom row gives us the quotient, according to the division algorithm
2x^4-5x^3-11x^2+20x+12=(x-2)(x-3)(2x^2+5x+2)
h. We can now factor the quadratic and get (x-2)(x-3)(2x+1)(x+2)
i. This gives us the four zeros: _____, _____, _____ and _____
3. Solving polynomial equations such as 6×4 + 13×3 + 61×2 + 8x – 10 = 0 is equivalent to finding the ___________ of the polynomial 6×4 + 13×3 + 61×2 + 8x – 10.
C. Complex Conjugate Pairs
1. If a + bi is a zero of a polynomial function with real coefficients, then the complex conjugate ______________ is also a zero.
a. If 2 – 3i is a zero of p(x)=x^4-4x^3+12x^2+4x-13, then ___________ is also a zero.
2. Find a third-degree polynomial function with real coefficients such that 3 + i and 5 are zeros.
a. Since the polynomial has real coefficients and 3 + i is a zero, ____________ is also a zero.
b. The three zeros are: _________, __________, and __________.
c. According to the factor theorem, three factors are: _____________, ___________, and _____________.
d. We can express the polynomial function as f(x) = a(x – [3 + i])(x – [3 – i])(x – 5) with a as any constant we choose.
e. If we choose a to equal 1 (which makes it the easiest) and multiply the first two factors together, we get f(x)=(x2 – 6x + 10)(x – 5).
f. If we multiply the polynomials we get f(x) = ___________________________.
3. Every odd degree polynomial with real coefficients has at least _______real zero(s).
D. According to the Intermediate Value Theorem, if f(3) and f(8) have opposite signs, then there exists at least one real zero between _____ and _____.
VII. Rational Functions
A. A rational function is a function in the form f(x) = (g(x))/(h(x)) , where g and h are __________________ functions and h(x) ≠ 0.
B. Domain and Intercepts
1. The domain of a rational function is all real numbers except those for which the denominator equals _______.
2. If f(x) has a y-intercept, it can be found by evaluating _______.
3. To find the x-intercepts of the function f(x) = (g(x))/(h(x)) , we solve the equation _______ = 0.
4. Determine the domain and intercepts of the function f(x) = (x+5)/(x^2-3x-18).
a. The domain is all real numbers except ______________.
b. The y-intercept is _______.
c. The x-intercept(s) is(are) __________.
C. Vertical Asymptotes
1. A rational function will have a vertical asymptote x = a, if a is a ____________ of the denominator (provided the numerator and denominator have no common _____________.)
2. Identify the vertical asymptotes in the function f(x) = (x+5)/(x^2-3x-18)
x = _______ and x = _________
D. Horizontal Asymptotes
1. Although a rational function can have many vertical asymptotes, it can have at most _________ horizontal asymptote.
2. The graph of a rational function will never intersect a _________________ asymptote but may intersect a __________________ asymptote.
3. A reduced rational function will have a horizontal asymptote whenever the degree of the numerator is ________________ than or equal to the degree of the denominator.
4. Let n be the degree of the numerator and d the degree of the denominator:
a. If n < d, then _____________ is the horizontal asymptote.
b. If n = d, then the horizontal asymptote is y = (the ratio of the leading _________________).
c. If __________, then there are no horizontal asymptotes.
E. Using Transformations to Sketch Rational Functions
1. Use transformations to sketch the graph of f(x)=(-0.45)/(x-2)-5.
a. Horizontally shift the graph of y = 1/x to the __________ 2 units to obtain the graph of y = 1/(x-2).
b. Vertically _________________ the graph of y = 1/(x-2) by a factor of 0.45 to obtain the graph of y=0.45/(x-2) .
c. Reflect the graph of y=0.45/(x-2), about the ____________ to obtain the graph of y=(-0.45)/(x-2) .
d. Vertically shift the graph of y=(-0.45)/(x-2) , ________ 5 units to obtain the graph of y=(-0.45)/(x-2)-5 .
F. Removable Discontinuities
1. For a rational function to have a removable discontinuity, the numerator and denominator must share a common ______________.
2. Consider the function (x)=(x^2-4)/(x+2) .
a. If we factor the numerator we see there is a common factor. f(x)=((x+2)(x-2))/(x+2) .
b. This function is not defined for x = ________.
c. But if we reduce the fraction we get f(x) = x – 2.
d. We can easily graph this function as a straight line. However, since x ≠ -2 for the function f, we “remove” the point on the line for which x = _____. This produces a “hole” on the graph.
G. Slant Asymptotes
1. If the degree of the numerator of a rational function is one more than the degree of the denominator, there will be a ______________ asymptote.
2. To find the equation of the slant asymptote, we divide the numerator by the __________________. This will give us a linear quotient, ax + b and a remainder. If we disregard the remainder, the line y = ax + b is the slant asymptote.
3. Find the slant asymptote of the function (x)=(3x^2-20x+13)/(x-7) .
a. Using synthetic division to divide the numerator by the denominator we get a quotient of 3x + 1 and a remainder of 20.
b. The slant asymptote is y = _____________.
H. Steps for Graphing Rational Functions
1. Find the ______________.
2. Cancel common factors determining the x-coordinates of any __________________ discontinuities.
3. Check for symmetry.
4. Find the y-intercept by evaluating __________.
5. Find the x-intercepts by finding the zeros of the _______________________.
6. Find the vertical asymptotes by finding the zeros of the ___________________. Use text values to determine the behavior on each side of the asymptotes.
7. Find any __________________ or ________________ asymptotes.
8. Plot points between each x-intercept and vertical asymptotes.
9. Complete the sketch.
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