Download presentation

Published byStanley Carson Modified over 6 years ago

1
**Quadratic Functions and their graphs Lesson 1.7**

A quadratic function- f(x) = ax2 + bx + c where a=/ 0 The graph of a quadratic function is called a parabola Two very special parts of a parabola: Vertex: The ‘turning’ point It is either a maximum or minimum. Axis of symmetry: A vertical Line that passes through the Vertex.

2
**The axis of symmetry can always be found by**

calculating: x = - (b) 2a Vertex: ( - b , f( - b) ) 2a a Discriminant: If b2 – 4ac > 0 Parabola crosses x-axis twice. There will be two x-intercepts. b2 – 4ac = 0 Parabola is ‘tangent’ to x-axis. There is only one x-intercept. b2 – 4ac < 0 Parabola never crosses the x-axis. No x-intercepts.

3
**The axis of symmetry is a vertical line midway between **

the x-intercepts. Therefore it is the ‘average’ of the x-intercepts. Example: Find the intercepts , the axis of symmetry , and the vertex , of this parabola. y = (x+4)(2x – 3) To find x-intercepts: Replace y with 0. 0 = (x+4)(2x – 3) Set each factor = 0 x + 4 = & 2x – 3 = 0 x = x = 3/2 Since the x-intercepts have already been found Find the average of these to find the axis. x = /2 = = -1.25 Now Vertex = (-1.25, f(-1.25)) (-1.25, ) Sketch the graph

4
**Example: Sketch the graph of the parabola. Label the **

intercepts, the axis of symmetry, and the vertex. y = 2x2 – 8x + 5 1st find the Axis of Symmetry X = -(-8) = 8 = 2 2(2) Now find the vertex: V = (2, f(2)) = (2,- 3) On this one, since b2 – 4ac = 44 , Which is not a perfect square, this can- Not be factored use the quadratic Formula to find x intercepts. X = 0.78 & 3.22 If you remember ‘c’ is always the y-intercept Sooo y-int = 5. Draw the ‘vertical axis’ at x = 2, Plot the vertex at (2,-3) Estimate the x-intercepts at 0.78 and 3.33, Plot the y-intercept at y = 5 and plot its symmetry point at (4,5) and sketch the parabola! If the equation can be written in the form of : y = a(x – h)2 + k vertex -- (h,k) axis of sym x = ‘h’

5
**If the equation can be written in the form of : **

y = a(x – h)2 + k vertex -- (h,k) axis of sym x = ‘h’ Example: a) Find the vertex of the parabola y = - 2x2 + 12x + 4 by completing the square. y = -2x2 + 12x + 4 = st subtract 4 from both sides y – 4 = -2x2 + 12x factor out a – 2 from both ‘x’terms y – 4 = -2(x2 – 6x ) complete the square inside the parentheses b = -6/2 = (-3)2 = 9 now add 9 inside the ( ) but add -2(9) or -18 to other side y – 4 – 18 = -2(x2 -6x +9) change to this look y – 22 = -2(x – 3) Now add 22 back to the right side y = -2(x – 3) line up y = a(x – h)2 + k y = a(x – h)2 + k Identify h = 3, k = 22 so vertex = (h,k) = (3,22)

6
**Example: b) Find the ‘x’ and ‘y’ – intercepts. **

y – intercept can be found from the given equation ‘c’ = y –intercept so y-intercept = 4 to find x-intercepts: let y = 0 and get: 0 = -2(x – 3)2 + 22 - 22 = -2(x – 3)2 11 = (x – 3)2 + √11 = x – 3 3 + √11 = x Example: Find the equation of the quadratic function ‘’f’ with f(-1) = - 7 and a maximum value of f(2) = -1 f(-1) = (-1,-7) A Maximum value at f(2) = -1 means (2,-1) is the vertex (h,k)

7
**so using h = 2 and k = - 1 gives us this working format:**

y = a(x – 2)2 + (-1) using the other point given (-1,-7) for x and y gives us: - 7 = a(-1 – 2) solve for ‘a’ - 6 = a(-3)2 - 6 = 9a divide by 9 - ⅔ = a So putting it all together y = -⅔(x – 2)2 - 1 b) Show that the function ‘f’ has no x-intercepts. The parabola must open downward since a = - ⅔ and since the vertex is located below the x-axis at (2,-1) it cannot cross the x-axis!

8
**Example: Where does the line y = 2x + 5 **

intersect the parabola y = 8 – x2 Check out the solution for example 3 found on page 40! (Show both the algebraic and graphical approach)

9
**Example: Find an equation of the function whose**

graph is a parabola with x-intercepts ‘1’ and ‘4’ and a y-intercept of ‘- 8’ Look over example 4 found on page 40. Homework: page 40 CE #1-6 all; page 41 WE #1-25 left column

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google