Quadratic Functions (Sections. and.) Definition. Quadratic functions are functions that can be written in the form where a, b, andc are real numbers. f(x) =ax + bx + c, a= 0 Example. Use a table to sketch the graph of f(x) =x. Now use your graphing calculator to graph each of the following and describe how it is di erent from the graph of f(x) =x : y =(x ) y = x y =x y =(x +) y = x + y = x y =(x ) y = x y =x y =(x +) y = x + y = x For c>0: y =(x c) y = x c y = cx y =(x + c) y = x + c y = cx Also: y = x
Example. Describe how the graph of g(x) couldbeobtainedfromthegraphoff(x) =x. g(x) =(x ) g(x) =(x +) g(x) =(x +) + g(x) =(x +) g(x) = (x ) + g(x) = (x ) + Remark. Every quadratic function (i.e. function of the form f(x) =ax + bx + c where a = 0), can be written in the form: f(x) =a(x h) + k. This is called the for a quadratic function and (h, k) is the.
0 Example. Given the graph of y = x (dashed lines) and the graph of y = f(x), determine a formula for f(x).
Definition. () Afunctionf is called increasing on an open interval I, ifforalla, b I, a<bimplies f(a) <f(b). (Rises as you go to the right) () Afunctionf is called decreasing on an open interval I, ifforalla, b I, a<bimplies f(a) >f(b). (Falls as you go to the right) () Afunctionf is called constant on I, iff(a) =f(b) foralla, b I. Example. For each of the functions below, determine the intervals of increase and decrease. f(x) =(x +) Increasing: g(x) = (x ) Increasing: Decreasing: Decreasing:
f(x) = x + Increasing: f(x) = (x +) + Increasing: Decreasing: Decreasing: Completing the Square If every quadratic function f(x) = ax + bx + c where a = 0 can be written in the form f(x) =a(x h) + k, howdoyougetitinthatform? f(x) =x +x Method
Example. Complete the square to write each quadratic function in the form f(x) = (x h) + k. Determine the vertex of the parabola, the axis of symmetry, and the intervals of increase/decrease. f(x) =x x + f(x) =x +x + Remark. Notice that the coordinates of the vertex for f(x) =x + bx + c are. Example. Write the following quadratic functions in the form f(x) =(x h) + k. f(x) =x +x f(x) =x +x +
What happens if a = 0inf(x) =ax + bx + c? How do you complete the square to write it in the form f(x) =a(x h) + k? f(x) =x +x Method Example. Complete the square to write each quadratic function in the form f(x) = (x h) + k. Determine the vertex of the parabola, the axis of symmetry, and the intervals of increase/decrease. f(x) = x +x f(x) =x x + Remark. Notice that the coordinates of the vertex for are. f(x) =ax + bx + c
Example. Write the following quadratic functions in the form f(x) =a(x h) + k. f(x) = x x f(x) =x +x + Definition. Let f(x) beafunction.thenf(c) is...... called a relative (local) maximum if there exists an open interval I containing c such that f(c) >f(x), for all x in I where x = c.... called a relative (local) minimum if there exists an open interval I containing c such that f(c) <f(x), for all x in I where x = c.
Remark. A function can have several relative maximums and several relative minimums. Here are some examples: Maximum and Minimum Applications (Section.) Example. A model rocket is launched into the air with an initial velocity of 0 ft/sec from the top of a building that is 0 feet high. The rocket s height, in feet, t seconds after it has been launched is given by the function: s(t) = t +0t +0. Determine the maximum height that the rocket will reach along with the time when the rocket will reach that maximum height.
Example. A manufacturer of dog houses has determined that when x hundred wooden doghouses are built, the average cost (in hundreds of dollars) per doghouse is given by C(x) =0.x 0.x +.. How many doghouses should be built in order to minimize the average cost per doghouse? Example. In business: P (x) = R(x) C(x) " " " Profit Revenue Cost (in $) (in $) (in $) where x is the number of units sold. Find the maximum profit, along with how many units must be sold when: R(x) =0 0.x C(x) =x +
Example. When Mentos are added to Diet Coke, the Diet Coke is propelled upward from ground level at a rate of feet per second. The height of the Diet Coke (in feet) after t seconds is given by the function s(t) = t +t. Determine the maximum height of the Diet Coke. Example. A landscaper has enough stone to enclose a rectangular koi pond next to an existing garden wall of a house with feet of stone wall. If the garden wall forms one side of the rectangle, what is the maximum area that the landscaper can enclose? What dimensions should the koi pond be?
Quadratic Equations (Section.) Definition. A quadratic equation is an equation that can be written in the form ax + bx + c =0 where a = 0. Example. Determine if the following equations are linear, quadratic, or neither. x = x p x +=x x = x x + x =x x = x = x How do you solve quadratic equations? Step # : Move everything to one side of the equation. Step # :Trytofactor. Step # :Ifyoucan tseemtofactortheequation,usethequadraticformula.
0 The solutions of ax + bx + c =0wherea = 0aregivenby: x = b ± p b ac. a Where does the quadratic formula come from?
Example. Solve the following equations. x x =0 x =x x = 0 y +y =
Example. (Continued from previous page) Solve the following equations. w +=w x x +=0 How many solutions does ax + bx + c =0havewherea = 0? Real Solutions and the Discriminant Overall Solutions
Applications with Quadratic Equations (Section.) Example. The number of U.S. forces in Afghanistan decreased to approximately,000 in 0 from a high of about 0,000 in 0. The amount of U.S. funding for Afghan security forces also decreased during this period. The function f(x) =.x +.x +. can be used to estimate the amount of U.S. funding for Afghan security forces, in billions of dollars, x years after 00. In what year was the amount of U.S. funding for Arghan security forces about $. billion? (Source: U.S. Department of Defense; Brookings Institution; International Security Assistance Force; ESRI ). Example. Astoneisthrowndirectlyupwardfromaheightof0feetwithaninitialvelocityof 0 feet per second. The height of the stone, in feet, t seconds after it has been thrown is given by the function s(t) = t +0t +0. Determine the total number of seconds the stone was in the air.
Quadratic Type Equations (Section.) Sometimes an equation is not a quadratic equation, but using a substitution of variables we can make it look like a quadratic equation. Consider: x x +=0 Example. Solve the following quadratic type equations. x x +=0 x x =
Polynomial Functions and Graphs (Section.-.) Definition. A polynomial function is a function of the form where the coe cients a n,a n,...a,a 0 are real numbers and the exponents are whole numbers. The degree of the polynomial P (x) isn, andtheleading coe cient of P (x) isa n. Example. Determine the degree, leading coe function. cient, domain, and range of each polynomial Function Degree Leading Domain Range Tiny Coe cient Sketch f(x) = f(x) =x + f(x) =x x + f(x) = x +x f(x) =x +x f(x) = x +x f(x) =x +x + x Can you describe any aspects of the shape of the graph of f(x) basedonitsdegree? Even Degree Odd Degree
Remark (The Leading-Term Test). For the polynomial P (x) =a n x n + a n x n + + a x + a 0 the behavior of the graph as x!(as the graph goes to the far right) and as x! the graph goes to the far left): (as n a n > 0 a n < 0 Example. Determine the end behavior of each polynomial. f(x) =x x + f(x) = x +x x + x f(x) =x x +x +x f(x) = x + x + x x f(x) = x x + x + f(x) =x 0 x +x 0
Solving Polynomial Equations (Sections.-.) Example. Solve the following equations. x x x =0 x +x = 0 But what if we couldn t see how to factor easily? If we can guess/find one root (i.e. solution) to the equation, we can use that to help us factor using the following relationship: x = c is a solution to f(x) =0() (x c) isafactoroff(x) To use this relationship, we need to know how to do polynomial long division. Example. x =isazerooff(x) =x x x +. So(x ) is a factor:
Example. Use polynomial long division to factor f(x), given a zero of f(x). f(x) =x +x x, x = f(x) =x x + x +x 0, x = Polynomial long division takes up a lot of space... and since we are only dividing by (x c) we can use a process called synthetic division that works in these cases. Example. Let s consider the problem f(x) =x x + x +x 0 with the zero x =from the previous example.
Example. Use synthetic division to factor the following functions, given a zero of the function. f(x) =x x x +,x = f(x) =x x +x +x, x = How can we use this to sketch a graph of a polynomial? Consider f(x) =x x x + (from the previous example). Step #: Find a zero by guess & check. Step #: Use synthetic division to factor. Step #: Completelyfactorf(x). Step #: Plotallx-intercepts. Step #: Use a sign chart to sketch the graph.
0 Example. Sketch the graph of each function. f(x) =x x + f(x) = x +x +x x
Polynomial Inequalities (Section.) Example. Solve the following inequalities. x x > 0 x +x apple x + x <x x x + x
Radical Functions (Section.) Definition. Radical functions are functions that involve p Example. Use a table to sketch the graph of f(x) = p x. n Now use your graphing calculator to graph each of the following and describe how it is di erent from the graph of f(x) = p x: y = p x y = p x y = p x y = p x + y = p x + y = p x y = p x y = p x Remark. So just like we saw for f(x) =x... for c 0 y = p x c y = p x c y = c p x y = p x + c y = p x + c y = c p x y = p x y = p x
Example. Describe how the graph of g(x) couldbeobtainedfromthegraphoff(x) =x. g(x) = p x + g(x) = p x + g(x) = p x g(x) = p x g(x) = p x + g(x) =0. p x In general, starting from the graph of y = f(x)... y = f(x c) y = f(x) c y = cf(x) y = f(x + c) y = f(x)+c y = cf(x) y = f(x) y = f( x) Example. Given the graph of y = f(x), sketch the graph of each. y =f(x ) y = f(x)+ y = f( x)
Example. Determine the domain and range for each of the following functions below. Recall: Thedomainandrangeoff(x) = p x are f(x) = p x + f(x) = p x + f(x) = p x + f(x) = p x + f(x) = p x f(x) = p x +
Example. Simplify the following. Solving Radical Equations (Section.) p p p p p p How do you use your calculator to help with things like p? Example. Use your calculator to round the following to three decimal places. p p p
How do we solve an equation like p x =? Example. Solve each of the following equations. p x += p y +=0 p x += p x +=
How do we solve an equation like p x ++=x? Example. Solve each of the following equations. p x +=x + p x +=x