m3u6d6 warm-up: find the zeroes: 1.x 2 – 6x – 16 = 0 2.2x 2 + 7x + 5 = 0 (x-8)(x+2)=0 x-8=0 or...
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M3U6D6 Warm-up:
Find the zeroes:1. x2 – 6x – 16 = 0
2. 2x2 + 7x + 5 = 0
(x-8)(x+2)=0x-8=0 or x+2=0 x=8 or x=-2
(2x+5)(x+1)=02x+5=0 or x+1=0 2x=-5 or x=-1 x=-5/2
M1U6D6 Axis of Symmetry and Graphing Quadratics
Objective:
To explore and graph quadratic functions.
Quadratic functions of the forms y = ax2 and y = ax2 + c
DEFINITIONS….
Standard Form of a Quadratic Function: A quadratic function is a function that can be written in the form y = ax2 + bx + c, where a ≠ 0. This form is called the STANDARD FORM OF A QUADRATIC FUNCTION.
Parabola: a “U-Shaped curve”
Axis of symmetry: The fold or line that divides the parabola into two matching halves.
Exploring Quadratic Graphs
Vertex:
If a > 0 in If a < 0 in
y = ax2 + bx + c y = ax2 + bx + c
then then
vertex is a minimum vertex is a maximum
Exploring Quadratic Graphs
Identify the vertex of each graph. Tell whether it is a minimum or a
maximum.
Exploring Quadratic Graphs
a.
The vertex is (1, 2).
b.
The vertex is (2, –4).
It is a maximum. It is a minimum.
Use the graphs below. Order the quadratic functions
(x) = –x2, (x) = –3x2, and (x) = x2 from widest to narrowest graph.
So, the order from widest to narrowest is (x) = x2, (x) = –x2,
(x) = –3x2.
12
12
(x) = –x2 (x) = x2 12
Of the three graphs, (x) = x2 is the widest and (x) = –3x2 is the narrowest.
12
(x) = –3x2
Exploring Quadratic Graphs
Reminder…..
+a in y = ax2 + bx + c makes it open up.
-a in y = ax2 + bx + c makes it open down.
If |a| is “small” (i.e. between 0 and 1) the graph is wide.
If |a| is “big” (i.e. greater than 1) the graph is narrow.
PROPERTY:
Graph of a Quadratic Function: The graph of y = ax2 + bx + c, where a ≠ 0, has the line x = -b/2a as its axis of symmetry. The x-coordinate of the vertex is –b/2a.
Graph the function y = 2x2 + 4x – 3.
Step 1: Find the equation of the axis of symmetry and thecoordinates of the vertex.
Find the equation of the axis of symmetry.x =b
2a– =
–42(2) = – 1
The x-coordinate of the vertex is –1.
y = 2x2 + 4x – 3
y = 2(–1)2 + 4(–1) – 3
= –5
To find the y-coordinate of the vertex, substitute –1 for x.
The vertex is (–1, –5).
Step 2: Find two other points.
Use the y-intercept.
For x = 0, y = –3, so one point is (0, –3).
Choose a value for x on the same side of the vertex.
Let x = 1
y = 2(1)2 + 4(1) – 3
= 3
For x = 1, y = 3, so another point is (1, 3).
Find the y-coordinate for x = 1.
Step 3: Reflect (0, –3) and (1, 3) across the axis of symmetry to get two more points.
Then draw the parabola.
Aerial fireworks carry “stars,” which are made of a sparkler-like material, upward, ignite them, and project them into the air in fireworks displays. Suppose a particular star is projected from an aerial firework at a starting height of 610 ft with an initial upward velocity of 88 ft/s. How long will it take for the star to reach its maximum height? How far above the ground will it be?
The equation h = –16t2 + 88t + 610 gives the height of the star h in feet at time t in seconds.
Step 2: Find the h-coordinate of the vertex.h = –16(2.75)2 + 88(2.75) + 610 Substitute 2.75 for t.h = 731 Simplify using a calculator.
The maximum height of the star will be about 731 ft.
Step 1: Find the x-coordinate of the vertex.
After 2.75 seconds, the star will be at its greatest height.
b2a– =
–882(–16) = 2.75