Quadratic Functions. Full Set of Notes. No Solutions

Transcription

1 Quadratic Functions Full Set of Notes No Solutions Graphing Quadratic Functions The graph of a quadratic function is called a parabola. Applications of Parabolas: Example 1: Use your graphing calculator to sketch the graph of y = x 2-4x+1 a) state the vertex b) state the equation of the axis of symmetry c) determine the maximum or minimum value of the function d) state the domain e) state the range f) calculate the y intercept g) calculate the x intercepts 1

2 Example 2: Identify the quadratic functions. Give a reason why the others are not quadratic functions. Example 3: Solve this quadratic equation by factoring 2x 2 +5x-3=0 Example 4: Solve using technology. -x 2 +3x+2=0 2

3 Example 5: The arch of a doorway can be modelled using a quadratic function of the form y=-2x 2 +6x. a) How high is the arch at the centre? b) How high is the arch 0.75 m from the right? c) How wide is the arch at the base? d) Where is the arch 2m high? Quadratic Regression Quadratic regression is the process you use with your graphing calculator to determine the quadratic function whose graph passes through 3 points on a parabola. Example 1: A doorway arch is 2m wide and 4m high in the middle. a) How high is the doorway 30cm from each side? b) How far from the middle is the doorway 3m above ground? 3

4 Example 2: The store manager knows that 240 books will be sold if the price per book is $12. He also knows that 60 fewer books will be sold each time the price increases by $0.80. Calculate the best price per book and the maximum revenue. Example 3: The perimeter of a rectangular field is to be 800 metres. Calculate the dimensions of such a field having a maximum area. 4

5 Graphing from the Standard Form y = a (x - p) 2 + q Sketch the graph of y = (x - 2) 2-3 Sketch the graph of y = -2 (x + 1)

6 The Parameters p : - p (notice the sign change) determines the horizontal position of the vertex q : q (notice no sign change) determines the vertical position of the vertex a : a determines the shape (steepness) of the graph if a is positive the graph opens upwards if a is negative the graph opens downwards Transformations 1. Start with the graph of y = 3(x - 2) translate the graph 4 units to the left and 1 unit up - make the graph twice as steep State the new equation. 2. Start with the graph of y = 8(x + 3) translate the graph 5 units to the right and 3 units down - make the graph only 1/4 as steep - reflect the graph upside down State the new equation 6

7 3. Start with the equation y = -2(x - 4) State the equation of another parabola whose translated 2 up and 3 to the left that is also congruent to the parabola y = -5(x +1 ) Forming Quadratic Equations use the standard form y = a(x - p) 2 + q insert the vertex, then use the extra piece of information to find the value of "a" 1. Find the equation of the parabola with vertex at (2, -3) passing through ( 5, 2). 7

8 2. Find the equation of the parabola with a maximum at (-2, 4) having a y intercept at A parabola has a minimum value of -6 and x intercepts of -1 and 9. Find its equation. 8

9 4. A parabola has an x intercept at -2 and a vertex at (2, -4). Find its equation. Review of Quadratic Functions 1. Three numbers are on a number line. The second number is 20 larger than the first number and the third number is 40 larger than the first number. Find the numbers if the sum of their squares is a minimum. 9

10 2. Sketch the graph of y = -(x + 3) General Form and Completing the Square The steps for completing the square - factor coefficient of x 2 - find half of coefficient of x - square and add - subtract - factor as a perfect square 1. Place into standard form and find the vertex. y = x 2 + 8x

11 The steps for completing the square - factor coefficient of x 2 - find half of coefficient of x - square and add - subtract - factor as a perfect square 2. Switch to standard form and find the range. y = x 2-6x - 2 The steps for completing the square - factor coefficient of x 2 - find half of coefficient of x - square and add - subtract - factor as a perfect square 3. Convert to standard form and find the axis of symmetry y = 2x 2-8x

12 The steps for completing the square - factor coefficient of x 2 - find half of coefficient of x - square and add - subtract - factor as a perfect square 4. Change to standard form and sketch y = -3x x + 5 The steps for completing the square - factor coefficient of x 2 - find half of coefficient of x - square and add - subtract - factor as a perfect square 5. Write in standard form and sketch. y = -x 2-4x

13 6. Write in general form y = -2(x - 3) Maximum and Minimum Problems 1. Two numbers have a difference of 10. Their product is a minimum. Determine the numbers. 13

14 2. The sum of two numbers is 8 and their product is a maximum. Determine the numbers. 3. A rectangular lot is bounded on one side by a river and on the other three sides by a total of 60 metres of fencing. Determine the dimensions of the largest possible lot. 14

15 4. When the price of a ticket to a movie is $8, 120 people will attend. It is known that every time the price goes up by $1, 10 fewer people attend. What ticket price would result in the greatest revenue? metres of fence material are to be used for the region shown below. Find the dimensions to enclose a maximum area. 15

16 computer games will be sold when the price is $20 per game. It is known that for every $5 increase in the price, there will be 20 fewer games sold. Find the price to yield a maximum revenue. Adjusting the Window Settings original widen to see x intercepts make higher to see top 16

17 Problems Example 1: It is known that 800 people attend a movie when the tickets cost $12 each. It is also known that 80 fewer people attend each time the ticket price goes up by $1. Find the ticket price giving the maximum revenue. Example 2: A rectangular field boarders on a river. 240 metres of fence material are needed on three sides to fence in some cattle. Calculate the dimensions of the field having a maximum area. 17

18 Example 4: One number is 40 larger than another number. Find both numbers if the sum of their squares is a minimum. Inverse of a Linear Function - to draw the inverse of a line switch the x and y coordinates of each point - to find the equation of the inverse switch x and y in the equation of the line 1. Draw the inverse of the line graphed below. 18

19 2. Draw the inverse of this line. 3. Find the equation of the inverse of y = 2x + 4. Isolate the y. 19

20 The Inverse of a Quadratic Function 1. Draw the inverse of the parabola shown below. 2. Draw the inverse of this parabola. 20

21 3. Find the equation of the inverse of y = x Sketch on your calculator. Review : Quadratic Functions 1. Given y = -3 x x a) general form or standard form? b) find the vertex c) find the range d) find the axis of symmetry 21

22 e) sketch the graph f) graph the inverse g) find the equation of the inverse 2. A parabola has x intercepts at x = -2 and x = 8 with a minimum of y = -4. a) find the equation using regression b) find the equation using algebra 22

23 3. Find the equation of the parabola with vertex at (2, 1) passing through (-1, 4). 4. An arch is 8 m wide at the bottom and 4 m high in the middle. How far from the center is the arch 2 m high? 23

24 5. Two numbers are on a number line. The second number is 15 more than the first number. Find the numbers if 4 times the square of the first number plus 2 times the square of the second number is a minimum. 24