MAT 105 SPRING 2009 Quadratic Equations Chapter 7 Quadratic Equations

A quadratic equation is one that can be written in the form _____________________________ where a, b, c are real numbers and a ≠0. The degree of a quadratic equation is ______. E.g.

a) ___________________________________ MAT 105 SPRING 2009 The related function has equation ______________ It has a graph in the shape of a _______________. Every quadratic equation has *two solutions (roots). They may be: a) ___________________________________ b) ___________________________________ c) ___________________________________

Methods of Solving Quadratic Equations With equation in form ax2 + bx + c =0 Graphing: Graph related function y = ax2 + bx + c and locate its real roots (x-intercepts) On TI-83/84, use 2nd Calc 2: Zero Factoring: If possible, factor the expression. Set each factor equal to zero and solve. Quadratic Formula:

Solve by graphing. Give answers to nearest tenth.

Solve by factoring.

Solve by factoring.

Solve by factoring.

Solve the higher order equation by factoring.

Solve the higher order equation by factoring.

Solve using the Quadratic Formula.

Solve using the Quadratic Formula (to two decimal places).

For equations in the form ax2 + bx + c =0, the discriminant is the value of ______________ . (The expression under the radical in the Quadratic Formula.) We can use the discriminant to determine the character (number and type) of the roots of a quadratic equation.

Character of the Roots If b2 – 4ac > 0 and is a perfect square, the equation has ________________________________________. If b2 – 4ac > 0 and is NOT a perfect square, the equation has _______________________________________. If b2 – 4ac = 0, the equation has __________________________________________. If b2 – 4ac < 0, the equation has ________________ ___________________________.

Describe the number and type of roots. MAT 105 SPRING 2009 Describe the number and type of roots.

Mathematical Modeling In real world applications we often encounter numerical data in the form of a table. The powerful mathematical tool, regression analysis, can be used to analyze numerical data. In general, regression analysis is a process for finding a function that best fits a set of data points. In the next example, we use a linear model obtained by using linear regression on a graphing calculator.

Regression Notes Regression: a process used to relate two quantitative variables. Independent variable: the x variable (or explanatory variable) Dependent variable: the y variable (or response variable) To interpret the scatterplot, identify the following: Form Direction (for linear models) Strength

Form Form: the function that best describes the relationship between the two variables. Some possible forms would be linear, quadratic, cubic, exponential, or logarithmic.

Direction Direction: a positive or negative direction can be found when looking at linear regression lines only. The direction is found by looking at the sign of the slope.

Strength Strength: how closely the points in the data are gathered around the form.

Making Predictions Predictions should only be made for values of x within the span of the x-values in the data set. Predictions made outside the data set are called extrapolations, which can be dangerous and ridiculous, thus extrapolating is not recommended. To make a prediction within the span of the x-values, hit  then . Next, arrow up or down  until the regression equation appears in the upper-left hand corner then type in the x-value and hit .

Example of Linear Regression MAT 105 SPRING 2009 Example of Linear Regression Prices for emerald-shaped diamonds taken from an on-line trader are given in the following table. Find the linear model that best fits this data. Weight (carats) Price 0.5 $1,677 0.6 $2,353 0.7 $2,718 0.8 $3,218 0.9 $3,982

MAT 105 SPRING 2009 Scatter Plots Enter these values into the lists in a graphing calculator as shown below .

MAT 105 SPRING 2009 Scatter Plots We can plot the data points in the previous example on a Cartesian coordinate plane, either by hand or using a graphing calculator. If we use the calculator, we obtain the following plot: Price of diamond (thousands) Weight (carats)

Example of Linear Regression (continued) MAT 105 SPRING 2009 Example of Linear Regression (continued) Based on the scatterplot, the data appears to be linearly correlated; thus, we can choose linear regression from the statistics menu, we obtain the second screen, which gives the equation of best fit. The linear equation of best fit is y = 5475x - 1042.9.

MAT 105 SPRING 2009 Scatter Plots We can plot the graph of our line of best fit on top of the scatter plot: Price of emerald (thousands) Weight (carats) y = 5475x - 1042.9

Making a Prediction Is it appropriate to use the model to predict the price of an emerald-shaped diamond that weighs 0.75 carats? If so, estimate the price. Is it appropriate to use the model to predict the price of an emerald-shaped diamond that weighs 2.7 carats? If so, estimate the price.

MAT 105 SPRING 2009 Quadratic Regression A visual inspection of the plot of a data set might indicate that a parabola would be a better model of the data than a straight line. In that case, rather than using linear regression to fit a linear model to the data, we would use quadratic regression on a graphing calculator to find the function of the form y = ax2 + bx + c that best fits the data. From the  CALC menu, choose 5: QuadReg

Example of Quadratic Regression MAT 105 SPRING 2009 Example of Quadratic Regression An automobile tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles.) x Mileage 28 45 30 52 32 55 34 51 36 47 Using quadratic regression on a graphing calculator, find the quadratic function that best fits the data. Round values to 6 decimal places.

Example of Quadratic Regression (continued) MAT 105 SPRING 2009 Example of Quadratic Regression (continued) Enter the data in a graphing calculator and obtain the lists below. Choose quadratic regression from the STAT Calc menu and obtain the coefficients as shown: This means that the equation that best fits the data is: y = -0.517857x2 + 33.292857x- 480.942857

Example of Quadratic Regression (continued) MAT 105 SPRING 2009 Example of Quadratic Regression (continued) Use the model to estimate the number of miles you could get from tires inflated at a) 35 psi and b) 40 psi.

Another Example of Modeling The following table shows crop yields, Y (in bushels), for various amounts of fertilizer used, x (in lbs/100 ft2), for 18 different equally sized plots. Plot # 1 2 3 4 5 6 7 8 9 19 11 12 13 14 15 16 17 18 x Fertilizer (lbs/ 100ft2) 10 20 25 30 35 40 Y Yield (bushels) 21 22

Example (continued) Use your calculator to graph a scatter plot of the data and comment on the type of relationship that exists between the two variables (the amount of fertilizer used , x, and the crop yield, y.) It appears that the data follows a quadratic relationship with a < 0.

Example (continued) Use the calculator to find the quadratic function of best fit. Give values to 4 significant digits. Sketch this function in the same window as your scatter plot.

Example (continued) Use the function to predict the optimal amount of fertilizer (in pounds per 100ft2) to use and the crop yield (in bushels) when the optimal amount of fertilizer is applied. Give values to 3 significant digits. Use the graphing calculator and the graph of the quadratic model to find the maximum point. According to the model, if we apply 31.5 pounds of fertilizer per 100 sq. feet, the crop yield will be 20.8 bushels.