Lesson 3.1 Vertices and Intercepts. Important Features of Parabolas

Transcription

1 Lesson 3.1 Vertices and Intercepts Name: _ Learning Objective: Students will be able to identify the vertex and intercepts of a parabola from its equation. CCSS.MATH.CONTENT.HSF.IF.C.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima. Important Features of Parabolas You can find the y-intercept by setting x equal to 0 in the equation for the parabola. You can find the vertex by rewriting the equation in vertex form. You can also find the x-coordinate of the b vertex by using the formula x. a Find the x-intercepts by setting y equal to 0 in the equation for the parabola. You can use factoring or the quadratic formula to solve the equation. 1

2 1. Find the vertices and intercepts of the parabolas graphed below. These are points in the coordinate plane, so your answers should be ordered pairs! a) b) Vertex: y-intercept: Vertex: y-intercept: c) d) Vertex: y-intercept: Vertex: y-intercept:

3 . Let s focus on how to use algebra to find the x and y intercepts from the equation of a parabola. Find the y-intercept by setting x equal to 0 and solving for y. Find the x-intercepts (if there are any), by setting y equal to 0 and solving for x. Directions: Find the x and y intercepts of the graphs below. Round your answers to the nearest hundredth. a) b) y-intercept: y-intercept: c) y-intercept: 3

4 3. Find the vertex from the parabola s equation. We can do this without rewriting the equation in vertex form. b Use the formula x to find the x-coordinate of the vertex. a Substitute for x in the parabola s equation to find the y-coordinate. a) y x x 6 11 b) y x x Vertex: Vertex: c) y x x 14 0 d) y x x 5 3 Vertex: Vertex: 4. Try this. A projectile was fired into the air from a Navy ship. The height, h, of the projectile depends on the time, t, since the object was launched into the air. The graph below shows the relationship between the object s height in feet and the amount of time in seconds since it was launched. Use the graph to answer the questions below. a)) When does the projectile reach its greatest height? b) What is the greatest height the projectile reaches? c) When will the projectile hit the ground? 4

5 5. Let s put it all together. a) f x x 4x 1 b) f x x x i) Find the vertex and plot it. i) Find the vertex and plot it. Vertex: Vertex: ii) Find other points around the vertex. ii) Find other points around the vertex. iii) Find the y-intercept. iii) Find the y-intercept. iv) Find the x-intercepts (if they exist). iv) Find the x-intercepts (if they exist). v) Graph the parabola! v) Graph the parabola! 5

6 Part : Projectile Motion Problems. When an object is thrown or shot into the air, its trajectory is a parabola and its height is a quadratic function of time given by the formula below: h 16t v0t h0, where v 0 is the object s initial velocity in feet per second and h 0 is its initial height in feet. 6. An object is launched directly upward at a 64 feet per second (ft/s) from a platform 80 feet high. a) Write an equation that can be used to calculate the height, h, of the object t seconds after it has been launched. b) How high will the object fly? When will it reach its maximum height? c) When will the object be 10 feet above the ground? d) When will the object hit the ground? 6

7 7. A rocket is fired vertically upwards into the air with an initial velocity of 80 ft/s from the top of an apartment building that is 35 feet high. How long will it take the rocket to reach the ground? 8. A ball is thrown in the air with an initial velocity of 14 m/s from an initial height of m. The ball s height h (in meters at time t (in seconds) can be modeled by the quadratic function reach a height of 1 m? h t 4.9t 14t. Does the ball ever Workspace: Explanation: 7

8 9. Desmos Exploration Quadratic Regression is a procedure for finding a parabola that comes as close as possible to the points of a scatterplot. It should be used when the points have the basic shape of a parabola. Desmos will do this for us. a) Find a quadratic model for the data below. x y Step 1: Open Desmos and create a table with these values. Step : Carefully type in this command: y ax bx c Step 3: Desmos will plot the parabola that comes as close as possible to all of the data points. It also gives the values for a, b, and c. Write the equation for the parabola Desmos created. Use the graph to find the vertex of the parabola. Use the graph to find the y-intercept. Are there any x-intercepts? If so, what are they? b) Try this one. x y Use quadratic regression to find a model for this data. Use the graph to find the vertex of the parabola. 8

9 10. Let s visit the real world!!! (source: 9

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