Chapter 6 Algebra *Linear and Nonlinear Functions *Graphing Quadratic Functions

12-1: Linear and Nonlinear Functions *Linear Function * Nonlinear Function - Straight lines indicating a constant rate of change. Curved line indicating that there is NOT a constant rate of change. Therefore, we can identify linear and nonlinear functions simply by looking at the graph! **Another method is to examine the equation of the function. Linear Nonlinear Slope-intercept form: y = mx +b Cannot be written in slope-intercept form Ex: y = x + 4 Slope tells us there is a constant rate of change! Ex: y = 6/x Since x is in the denominator, it cannot be written in m = 1 and b = 4 slope-intercept form.

Even a table can help us identify! X Y 20 5 16 10 12 15 8 X Y 2 4 20 6 54 8 104 Is the rate of change constant or is it increasing/decreasing at different rates? Let's Try... Determine whether each represents a linear or nonlinear function. 1. 2. y = 5x2 + 3 4. y – 4 = 5x y = -x + 3 y = 2 –x/2 + 1

Quadratic Function A function in which the greatest power of the variable is 2. The graph of a quadratic function will be a U-shaped curve, called a parabola. Ex: y = x2 y = 2x2 - 2x -8 y = -2x2 The graph represents the quadratic function: y = -2x2 Axis of symmetry – vertical line that passes through the vertex. Vertex – turn around point. Notice that the graph is opening downward. This is because of the -2. Whenever the coefficient of the x2 value is negative, the graph of the quadratic function will open down. This makes the y-coordinate of the vertex the maximum of the graph!! x -2x2 y (x, y) -2 -2(-2)2 -8 (-2,-8) -1 -2(-1)2 (-1,-2) -2(0)2 (0,0) 1 -2(1)2 (1,-2) 2 -2(2)2 (2,-8) When graphing quadratic equations, you will form a table by substituting values for x into the function, and then plotting ordered pairs.

Graph the following quadratic functions… y = 5x2 y = 3x2 + 1 y = -2x2 - 1 x y = 5x2 y (x, y) x y = 3x2 +1 y (x, y) x y = -2x2 - 1 y (x, y)

Formulas to keep in mind!!! Slope = y2 – y1 x2 – x1 Slope-intercept: y = mx + b, where m = slope & b = y-intercept Geometric Sequence: an = a1r (n – 1) Arithmetic Sequence: an = a1 + (n – 1)d Midpoint = x2 + x1 , y2 + y1 2 2 Review Surface Area formulas and Conversion

Practice Problems: What is the sum of the measures of <K and <L? Zack conducted a survey in his school. He asked each student to choose his or her favorite subject. Zack wants to create a graph showing the percentage of students who chose each subject as their favorite. Which type of graph would best represent these data? 47° 94 ° 133 ° 227 ° 133 ° K L J stem-and-leaf circle graph line graph scatterplot

What is the slope of the line passing through the points (1, 3) and (5, 0)? 4. Rob would like to buy a $20 CD that is on sale for 15% off. He estimates that he will save at least $4.00. *In your answer document, determine whether or not Rob’s estimate is reasonable. Show or explain how you decided if Rob’s estimate is reasonable. - 4/3 - ¾ ¾ 4/3

6. What is the value of x when 3x < -2x + 15? 5. Evaluate: (23) (3-2). 6. What is the value of x when 3x < -2x + 15? Kyle spins each spinner once and multiplies the two numbers he lands on. What is the reasonable prediction for what might happen? -36 - 6/6 8/9 72 X > 3 X < 3 X < 15 X > 15 3 2 5 6 2 4 5 1 3 The most likely product is 12. The most likely product is 10. The product cannot be less than 4. The product cannot be greater than 18.