0 1 4 1 (–1, 1) (0, 0) (1, 1) (2, 4) y x 4 (–2, 4) Axis of symmetry: x = 0 ( y=x 2 is symmetric with respect to the y-axis ) Vertex (0, 0) When a > 0 , the parabola opens upwards and is called concave up. The vertex is called a minimum point. y 2 1 0 – 1 – 2 x
a = 1 . What if a does not equal 1? Consider the equation y = – 4 x 2 . 0 – 4 – 4 (0, 0) – 16 y – 16 (–2, –16) (–1, –4) (1, –4) (–4, –16) x When a < 0 , the parabola opens downward and is called concave down. The vertex is a maximum point. What is a ? a = – 4 y 2 1 0 – 1 – 2 x
The graph of f ( x ) = ax 2 is a parabola with the vertex at the origin and the y axis as the line of symmetry.
If a is positive , the parabola opens upward , if a is negative , the parabola opens downward .
If a is greater than 1 ( a > 1), the parabola is narrower then the parabola f ( x ) = x 2 .
If a is between 0 and 1 (0 < a < 1), the parabola is wider than the parabola f ( x ) = x 2 .
5.
Graphing the parabola y = f ( x ) = ax 2 + k Algebraic Approach: y = – 4 x 2 – 3 Numerical Approach: Graphical Approach: Consider the equation y = – 4 x 2 – 3 . What is a ? a = – 4 x Vertex (0, -3) -16 -4 0 -4 -16 -4 x 2 -19 -7 -3 -7 -19 - 4 x 2 - 3 2 1 0 – 1 – 2 x
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y = – 4 x 2 – 3 . x Vertex (0, -3) y = – 4 x 2 In general the graph of y = ax 2 + k is the graph of y = ax 2 shifted vertically k units. If k > 0, the graph is shifted up. If k < 0, the graph is shifted down. (P. 267) The graph y = – 4 x 2 is shifted down 3 units.
7.
a = – 4. What effect does the 3 have on the function? y x y = – 4 x 2 y = – 4( x – 3) 2 Consider the equation y = – 4( x – 3) 2 . What is a ? The axis of symmetry is x = 3. Numerical Approach: Axis of symmetry is shifted 3 units to the right and becomes x = 3 -16 -4 0 -4 -16 – 4 x 2 2 1 0 – 1 – 2 x -4 -16 -36 -64 -100 - 4 ( x- 3) 2 -36 3 0
Each function we just looked at can be written in the form (x – h) 2 + k, where (h , k) is the vertex of the parabola, and x = h is its axis of symmetry.
(x – h) 2 + k – vertex form
x = 3 ( 3 , 0 ) y = (x – 3 ) 2 or y = (x – 3 ) 2 + 0 x = 0 ( 0 , 2 ) y = x 2 + 2 or y = (x – 0 ) 2 + 2 x = 0 ( 0 , 0 ) y = x 2 or y = (x – 0 ) 2 + 0 Axis of Symmetry Vertex Equation
This function can be rewritten as y = [x – (-2)] 2 + 1.
Then, h = -2 and k = 1
The vertex is at (h , k) = (-2 , 1), the axis of symmetry is x = -2. The graph is the same shape as the graph of y = x 2 , but is translate 2 units left and 1 unit up.
Now use the information to draw the graph.
Step 1 Plot the vertex (-2 , 1)
Step 2 Draw the axis of symmetry, x = -2.
Step 3 Find and plot two points on one side of the axis symmetry, such as (-1, 2) and (0 , 5).
How does the value of a in the general form y = a(x – h) 2 + k affect a parabola? Compare the graphs of the following functions to the parent function, y = x 2 .
a.
b.
c.
d.
y = x 2 a b d c
14.
Example: Write a quadratic function for a parabola with a vertex of (-2,1) that passes through the point (1,-1).
Since you know the vertex, use vertex form! y=a(x-h) 2 +k
Plug the vertex in for (h,k) and the other point in for (x,y). Then, solve for a.
-1=a(1-(-2)) 2 +1
-1=a(3) 2 +1
-2=9a
Now plug in a, h, & k!
15.
Standard 9 Write a quadratic function in vertex form Vertex form- Is a way of writing a quadratic equation that facilitates finding the vertex. y – k = a ( x – h ) 2
The h and the k represent the coordinates of the vertex in the form V( h, k).
The “a” if it is positive it will mean that our parabola opens upward and if negative it will open downward.
A small value for a will mean that our parabola is wider and vice versa.
16.
Standard 9 Write a quadratic function in vertex form Write y = x 2 – 10 x + 22 in vertex form. Then identify the vertex. y = x 2 – 10 x + 22 Write original function. y + ? = ( x 2 – 10 x + ? ) + 22 Prepare to complete the square. y + 25 = ( x 2 – 10 x + 25 ) + 22 y + 25 = ( x – 5) 2 + 22 Write x 2 – 10 x + 25 as a binomial squared. y + 3 = ( x – 5) 2 Write in vertex form. Add – 10 2 2 ( ) = (–5) 2 = 25 to each side. The vertex form of the function is y + 3 = ( x – 5) 2 . The vertex is (5, – 3). ANSWER
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EXAMPLE 1 Write a quadratic function in vertex form Write a quadratic function for the parabola shown. SOLUTION Use vertex form because the vertex is given. y – k = a ( x – h ) 2 Vertex form y = a ( x – 1) 2 – 2 Substitute 1 for h and –2 for k . Use the other given point, ( 3 , 2 ) , to find a . 2 = a ( 3 – 1) 2 – 2 Substitute 3 for x and 2 for y . 2 = 4 a – 2 Simplify coefficient of a . 1 = a Solve for a .
18.
EXAMPLE 1 Write a quadratic function in vertex form A quadratic function for the parabola is y = ( x – 1) 2 – 2 . ANSWER
19.
EXAMPLE 1 Graph a quadratic function in vertex form Graph y – 5 = – ( x + 2) 2 . SOLUTION STEP 1 STEP 2 Plot the vertex ( h , k ) = ( – 2, 5) and draw the axis of symmetry x = – 2. 14 Identify the constants a = – , h = – 2, and k = 5. Because a < 0, the parabola opens down. 14
20.
EXAMPLE 1 Graph a quadratic function in vertex form STEP 3 Evaluate the function for two values of x . Plot the points (0, 4) and (2, 1) and their reflections in the axis of symmetry. STEP 4 Draw a parabola through the plotted points. x = 0 : y = ( 0 + 2) 2 + 5 = 4 14 – x = 2 : y = ( 2 + 2) 2 + 5 = 1 14 –
21.
GUIDED PRACTICE for Examples 1 and 2 Graph the function. Label the vertex and axis of symmetry. 1. y = ( x + 2) 2 – 3 2. y = –( x + 1) 2 + 5
22.
GUIDED PRACTICE for Examples 1 and 2 3. f ( x ) = ( x – 3) 2 – 4 12