35.4 – Polynomial Functions A polynomial function is any function of the formThe degree of the polynomial is n, where n is the highest power of x.A polynomial of degree 3 is called a cubic function:A polynomial of degree 2 is called a quadratic function:A polynomial of degree 1 is called a linear function:A polynomial of degree 0 is called a constant function:
4EVERY polynomial function has a domain of all real numbers: Why EVERY polynomial function has a domain of all real numbers: Why? There are no values of x for which the function is not defined. In other words, we can evaluate the function for ANY value of x.The range of polynomial functions is easiest to find by considering the graph of the function.Many polynomial functions can just be graphed using the transformation techniques we learned in the previous section.
5Linear Functions The slope of the line that passes through the points and isGiven a point on a line and its slope m, the following Point-Slope Form of a Line can be used to find its equation.Slope-Intercept Form of a Line:where m is the slope of the line and b is the y-intercept.General Form of a Line: Ax+By+C=0, where A, B, and C are integers.
6Example: Find an equation of the line that passes through the points (2, -3) and (4, 6). Write your answer in slope- intercept form and also in general form.
7A horizontal line through the point (a,b) has equation y=b A horizontal line through the point (a,b) has equation y=b. Note that a horizontal line is really just a constant function.A vertical line through the point (a,b) has equation x=a. Note that this is not a function at all! It fails the Vertical Line Test.Example: Find equations of the horizontal and vertical lines that pass through the point (9, -2).
8Quadratic FunctionsWe know the graph of the quadratic function is a parabola. EVERY quadratic function has the shape of a parabola and can be written as a transformation of the parent function by writing it in standard form.The standard form a quadratic function isWhen written in standard form, the vertex of the quadratic can be easily identified as the point (h, k).If a>0, the parabola opens upward.If a<0, the parabola opens downward.If a quadratic function is given in the form , you can complete the square to write it in standard form.
9Write the following quadratic functions in standard form and identify the vertex of each parabola. Does the parabola open upward or downward?
10An alternate (easier) way to find the vertex is to use the following: Given a quadratic function the vertex (h, k) is given byExample: Find the vertices of the quadratic functions below. Do the quadratics open upward or downward?
11InterceptsTo determine the x-intercepts of any function, we want to know where the function crosses or touches the x- axis. This occurs when f(x)=0. Set the function equal to 0 and solve to find the x-intercepts.To determine the y-intercept, we want to know where the function crosses or touches the y-axis. This occurs when x=0. Evaluate the function at x=0 to determine the y-intercept (if one exists).
12Graph the same quadratic functions below Graph the same quadratic functions below. Identify and plot any intercepts.Range:_______________ Increasing on:________________ Decreasing on:________________ Is the vertex a max or min?________ Axis of symmetry:__________
13Graph the quadratic function below. Identify and plot any intercepts. Range:_______________ Increasing on:________________ Decreasing on:________________ Is the vertex a max or min?________ Axis of symmetry:__________
14Find the x and y-intercepts for the following functions Find the x and y-intercepts for the following functions. Create a sign chart as we did with inequalities to determine where the function is above and below the x-axis. Then, sketch a graph of each function.
165.5 – Rational FunctionsA rational function is a function of the form , where both f(x) and g(x) are polynomials and whereSince the denominator cannot be 0, the domain of a rational function consists of all values of x for which the denominator is not 0. In other words, find all values of x that make the denominator 0 and exclude them.Example: Find the domain of the function
17Notice that in the previous function, the numerator also had a factor of (x-3). Although these terms would cancel when simplifying the expression, since the original function is undefined at x=3, it is still not included in the domain.If a factor completely cancels out of the denominator when simplified, there is a “hole” in the graph of the function at this location. In the previous function, there would be a “hole” in the graph at x=3. To find the y-value of the “hole,” plug in x=3 into the simplified function:If a factor does NOT cancel from the denominator, there is a vertical asymptote at that location. In the previous function, the graph has vertical asymptotes x=-9 and x=0.
18A rational function will either increase to ∞ or decrease to -∞ without bound as it approaches a vertical asymptote on either side. A rational function will NEVER cross a vertical asymptote.Sketch the graph of a function f(x) with vertical asymptotes at x=-5 and x=5 with the following properties:f(x)∞ as x -5 from the leftf(x)-∞ as x -5 from the rightf(x)-∞ as x 5 from the leftf(x)-∞ as x 5 from the right
19Determine the domain, vertical asymptotes, and holes for the following functions.
20InterceptsJust as with polynomial functions, to find x-intercepts, set the function equal to 0. With a rational function, this can ONLY occur when the numerator is equal to 0 (and there is no hole there). Finding the x-intercepts of a function is sometimes referred to as finding the zeros of a function.To find the y-intercept, evaluate the function at x=0, if possible.Find the intercepts for the previous two functions.
21Horizontal Asymptotes Horizontal asymptotes are horizontal lines which the graph of the function approaches as x ∞ or x-∞. In other words, they describe the behavior of the graph at the extreme left and right of the domain.In the graph below, you can see that as x ∞ and as x -∞, the graph is approaching the horizontal asymptote y=1.
22Determining Horizontal Asymptotes Consider the rational functionNote that n is the degree of the numerator and m is the degree of the denominator.If n<m, then y=0 is the horizontal asymptote.If n=m, then y= is the horizontal asymptote.This is the ratio of the leading coefficients.If n>m, then there is no horizontal asymptote.When graphing a rational function, we can again use a sign chart as we did with rational inequalities to determine where the graph is positive and where it is negative.
23Determine the horizontal asymptotes of the following functions, if any.
24Find all intercepts, asymptotes, holes, and use a sign chart to help sketch the general shape of the function.
255.6 – Radical FunctionsWith functions that involve radicals, we know that we cannot take an even root (square root, fourth root, etc) of a negative number. So, to find the domain of a function that involves radicals, we must ensure that any expression inside an even root is ≥0.If, in addition, the function has a denominator, we must still exclude the values of x that make the denominator 0.Find the domains of the following functions:
29Things to Do Before Next Meeting: Work on Sections until you get all green bars!Write down any questions you have.Continue working on masteringMake sure you have taken the Chapter 6 Test before our next meeting.