Review for Exam #3
3.1 Quadratic Functions
A quadratic functions is any function of the form , where a, b, c are real numbers with . The graph of a quadratic function is a parabola.
If ,
The graph opens upward and has the minimum point, vertex. The vertex is . The vertex line is the axis of symmetry.
If ,
The graph opens downward and has the maximum point, vertex. The vertex is . The vertex line is the axis of symmetry.
The standard form of a quadratic function is , where the point is the vertex of the parabola.
*Graph Quadratic Functions
Step 1 If , it opens upward. If , it opens downward.
Step 2 The vertex of the parabola is (h, k) or .
Step 3 Find x-intercepts by solving .
Step 4 Find y-intercepts by computing .
Step 5 Plot the intercepts and the vertex. Connect these points with a smooth curve.
Example 1) Let .
a) Determine whether has a maximum or a minimum and find the maximum or minimum value of and where it occurs.
b) Graph the function and determine the function’s domain and range (in interval notation).
Example 2) A stone mason has 60 feet of stone wall to enclose a rectangular patio that borders a house. If the stone wall is not used along the side of the house, find:
a) The dimensions of that patio that yield the maximum area for the patio.
b) The maximum area that the mason may enclose.
Example 3) A projectile is shot vertically into the air from a platform. The height, , in feet, of the projectile above the ground after t seconds is modeled by . Give solutions accurate to three decimal places.
a) What is the height of the projectile above the ground after 2 seconds?
b) How many seconds will it take the projectile to reach its maximum height?
c) What is the maximum height of the projec
d) How long will it take the projectile to reach the ground?
3.2 Polynomial Functions and Their Graphs
is a polynomial function of degree n, where are real numbers with . The number is called the leading coefficient.
*Graphing a Polynomial Function
Step1 Use Leading Coefficient Test to determine the end behavior.
n: odd (Opposite behavior at each end)
If , the graph falls to the left and rises to the right.
If , the graph rises to the left and falls to the right.
n: even (Same behavior at each end)
If , the graph rises to the left and rises to the right.
If , the graph falls to the left and falls to the right.
Step2 Find x-intercepts by setting . For a factor of
a. If k is even, the graph touches the x-axis at r and turns around
b. If k is odd, the graph crosses the x-axis at r.
c. If k>1, the graph flattens out at (r, 0).
Step 3 Find y-intercept by computing .
Step 4 Use symmetry, if applicable, to help draw the graph:
a. y-axis symmetry:
b. Origin symmetry:
Step 5 Use the fact that the maximum number of turning points of the graph is to check whether it is drawn correctly.
Example 4) Graph
Example 5) Graph
Example 6) Use the graph of the polynomial below, to answer True or False for the statements that follow.
a) The sign of the leading coefficient is negative.
b) The degree of the polynomial is even.
c) At , has a zero of even multiplicity.
3.3 Dividing Polynomials; Remainder and Factor Theorem
*Long Division
Divide-Multiply-Subtract-Bring down
Example 7) Find the quotient and the remainder:.
*Synthetic Division
We can use synthetic division only if the divisor is in the form .
Example 8) Find the remainder after division by .
*The Remainder Theorem
If is divided by , the remainder is .
*The Factor Theorem
a. If , then is a factor of .
b. If is a factor of , then .
3.4 Zeros of Polynomial Functions
*Properties of Polynomial Equations
1. If is a polynomial of degree n, then the equation has n complex roots, possibly real and not necessarily distinct.
2. If is a root of a polynomial equation with real coefficient , then the complex imaginary number is also a root.
3. If are roots of , then with .
4. Possible rational zeros=.
*Finding Zeros of a Polynomial Function
Step 1 List all possible rational zeros
Step 2 Use synthetic division to find a rational zero among the possibilities.
Step 3 Factor the polynomial using synthetic division.
Step 4 Repeat Step 1 through Step 3 until you get a quadratic factor.
Example 9) Let.
a) List all possible rational zeros of .
b) Use synthetic division to find a zero of . Use that zero to algebraically find all the zeros of .
c) Write as a “product of linear factors”.
Example 10) Using long division and given root, find the solution set of the given polynomial equation. Give exact solutions.
Equation: Root: 2i
3.6 Polynomial and Rational Inequalities
*Solving Polynomial Inequalities
Step 1 Express the inequality in the form or .
Step 2 Solve the equation .
Step 3 Locate the boundary points on a number line and separate the line into intervals.
Step 4 Choose one test value within each interval and evaluate at that number.
Step 5 Write the solution set, selecting the interval or intervals that satisfy the given inequality.
*Solving Rational Inequalities
Step 1 Express the inequality so that one side is zero and the other side is a single quotient.
Step 2 Set the numerator and the denominator of equal to zero.
Step 3 Locate the boundary points on a number line and separate the line into intervals.
Step 4 Choose one test value within each interval and evaluate f at that number.
Step 5 Write the solution set, selecting the interval or intervals that satisfy the given inequality. (Never include the value that cause the denominator to equal zero)
Example 11) Solve and write the solution in interval notation.
Example 12) Solve and write the solution in interval notation.
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