4-1 (Part 2) Graphing Quadratics, Interpreting Parabolas

Transcription

1 4-1 (Part 2) Graphing Quadratics, Interpreting Parabolas Objectives Students will be able to: Find the vertex and y-intercept of a parabola Graph a parabola Use quadratic models to analyze problem situations. Part 1: Exploring Quadratics Most of the functions we ve seen so far have been linear. That is, their graphs form a line. However, not every function will form a line. Let s explore a new type of function called a quadratic. In your prep assignment, you saw that a quadratic function has a new term that includes an x 2. Consider the following quadratic function y = x 2 1 a) Fill in the table below by substituting the x-values into the equation, then graph the points onto the coordinate plane. Table Graph X Y b) Describe what you see in the graph. How would you describe the shape? Where is the lowest point on the graph? Are there any x or y-intercepts?

2 Part 2: Graphing Addiction Introduction 1 Prescriptions to highly addictive drugs are on the rise in the U.S. 2 This increase has led to a drastic increase in deaths due to opioids. 3 It also leads to more and more people being admitted for medical treatment due to addiction to opioids. (1) How many Americans, per 1,000,000, do you think were admitted for medical treatment due to opioid addiction in 2010? Take a guess. Data from National Vital Statistics 4 can be used to find a model for the number of Americans admitted for medical treatment due to opioid addiction. One model is: y = 0.25x x + 11 where x is the number of years since 2000 and y is the rate of admissions per 1,000,000 people. Using Desmos we created the following graph: 1 Image from

3 (2) Answer the following using that this model: A. Recall from the Prep Assignment that a quadratic has the general form y = ax 2 + bx + c. For the opioid model in part 1, what are the values of a, b, and c? B. From the graph, find the y-intercept. C. Interpret the meaning of the y-intercept. D. Use the opioid model graph to estimate the rate of admission in the year Write your answer as a number of people out of 1,000,000 and compare it to your prediction in #1. Plot that point on the graph on the previous page. E. In what year does it look like opioid admissions will reach 60 people per 1,000,000? F. You ll notice the model doesn t show the whole shape of the parabola, and only displays the right side of the curve. Why do you think the left side was omitted?

4 Skills Example Consider the quadratic equation: y = x 2 + 6x 16. The graph of this equation with all of the points/features listed above looks like this: The important points are: Vertex at (-3,-25) x-intercepts at (-8,0) and (2,0) y-intercept at (0,-16) This parabola is said to open upwards, with a minimum value at the vertex. However, some parabolas open downward and have a maximum value at the vertex, as we will see in a moment. To graph a parabola, we will follow these steps: 1. Find the vertex of the parabola. Find the x-coordinate using the formula x = b 2a Find the y-coordinate by substituting the x-coordinate into the equation 2. Determine if the parabola opens up or opens down. 3. Find the y-intercept of the parabola. 4. Use symmetry to find the point that is the mirror image of the y-intercept. 5. Plot all points and then sketch the graph, making sure the vertex is not "sharp" and the graph is symmetrical.

5 Example Graph the parabola y = 2x 2 + 8x 24 1.) Find the vertex Since a = 2, b = 8, c = 24 we have So the x coordinate of the vertex is x = 2. x = b 2a = (8) 2(2) = 8 4 = 2 Find the y-coordinate by substituting x = 2 into the equation: y = 2( 2) 2 + 8( 2) 24 Follow the order of operations, exponents, then multiplication, then addition/subtraction: So the vertex is at ( 2, 32). 2.) Determine if the parabola opens up or down: y = 2(4) + 8( 2) 24 y = y = 32 For y = 2x 2 + 8x 24, a = 2 > 0. This tells us the parabola opens up. 3.) Find the y-intercept: The y-intercept is the point (0,c). For y = 2x 2 + 8x 24, c = 24, so the y-intercept is (0, 24). We plot that as well. 4.) Use symmetry to find another point: If we draw the axis of symmetry, we notice that the y-intercept is two units to the right of the axis. So the point (0,-24) will have a mirror image point two units to the left of the axis. That is the point (-4,-24). We plot that point as well. 5.) Connect all of the points with a smooth curve.

6 p (Profit, $) Applications Maximizing Profits An airline charter company is trying to maximize its profits. Tickets are normally around $800, but the company reduces the fare by about $10 per ticket for each additional person in the group. The plane has a total of 90 seats, including 5 for required pilots and crew. The company has used all of this information to create the following model for their profit: p = 10n n 2000 where n is the number of people in the group (not including the pilots or crew), and p is the total profit for that trip, in dollars. (3) A. Draw a graph of this model. Use the process you saw in the Prep Assignment and practiced in the previous problem. Show all steps and work in your notes for studying later n (Size of charter group) B. If the company wanted to maximize profits for a trip, approximately what size group should the company target for that trip? How much is the maximum profit?

7 Just How High Can You Go? Physics rules most of the world around us: It governs our ability to walk (gravity), helps us construct multistory skyscrapers, and gives us the ability to fly in planes, among many other things. A common application from physics tells us what happens to an object when you throw an object straight up in the air and then let it fall to the ground. The general formula is: where: h = 16t 2 + vt + b h = the height (feet) of the object at a given time. t = the amount of time (seconds) since the object was thrown v = the initial velocity (feet per second) the object was thrown b = the beginning height (feet) from which the object was thrown (4) Read the information above and then answer the following: A. Find the simplified equation for the situation when an object is thrown from a beginning height of 80 feet off the ground at an initial velocity of 64 feet per second. Write your equation. B. Graph your equation on Desmos and include the important points. Sketch a graph of the model on the next page. Only draw the part of the graph that makes sense. C. What is the maximum height the object will reach? Verify this with both the graph and the simplified equation. D. Use the Desmos graph to determine when the object is 10 feet above the ground. E. Use the Desmos graph to determine when the object hits the ground.

8 h (height, feet) t (time, seconds)