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ALGEBRA 1
Lesson 9-2 Warm-Up
ALGEBRA 1
“Quadratic Functions” (9-2)
How do you find a parabola’s axis of symmetry (fold or line that divides the parabola into two matching halves).
Recall that the standard form of a quadratic function is ax2 + bx + c and that the graph of a quadratic function is a parabola (U-shaped curve). The position of a parabola’s axis of symmetry (fold or line that divides the parabola into two matching halves) is based on a ratio between the b and a values (the c value is the y-intercept). Since the parabola of a quadratic function open upward or downward, the axis of symmetry is a vertical line, which means it is somewhere along the x axis.
Rule: Axis of Symmetry: The axis of symmetry, x, of a function in the form of ax2 + bx + c is:
x (axis of symmetry) = – • = –
Examples: y = 2x2 + 2x
– = - = -
In the graph, notice that there is no
y-intercept, since there is no c value.
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ba
12
b2a
b2a
22 (2)
1 2
ALGEBRA 1
“Quadratic Functions” (9-2)
How do you graph a quadratic function?
To graph a quadratic function, you will need to find the vertex, the axis of symmetry, and at least two points on each side of the axis of symmetry.
Step 1: Find the axis of symmetry and the vertex.
The axis of symmetry is x = 1.
The vertex is (1, 8)
Step 2: Find two other points on one side of the line of symmetry (if possible, one should be the y-intercept)
When x = -1, y = -4, so another point is (-1, -4).
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ALGEBRA 1
“Quadratic Functions” (9-2)
Step 3: Reflect the points you found (graph the point that is the same distance on the opposite side of the line of symmetry)
ALGEBRA 1
Graph the function y = 2x2 + 4x – 3.
Step 1: Find the axis of symmetry and the coordinates of the vertex.
Find the equation of the axis of symmetry.x =b
2a– =
–42(2) = – 1
The axis of symmetry is x = –1.
The vertex is (–1, –5).
y = 2x2 + 4x – 3
To find the y-coordinate of the vertex, substitute –1 for x.
y = 2(–1)2 + 4(–1) – 3
= –5
Quadratic FunctionsLESSON 9-2
Additional Examples
ALGEBRA 1
(continued)
Step 2: Find two other points on the graph.
For x = 0, y = –3, so one point is (0, –3).
Use the y-intercept.
Choose a value for x on the same side of the vertex.
Let x = 1
Find the y-coordinate for x = 1.y = 2(1)2 + 4(1) – 3
= 3
For x = 1, y = 3, so another point is (1, 3).
Quadratic FunctionsLESSON 9-2
Additional Examples
ALGEBRA 1
(continued)
Step 3: Reflect (0, –3) and (1, 3) across the axis of symmetry to get two more points.
The domain is the set of all real numbers. The range is {y : y ≥ –5}.
Quadratic FunctionsLESSON 9-2
Additional Examples
Then draw the parabola.
ALGEBRA 1
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. Since the coefficient of t2 is negative, the curve opens downward, and the vertex is the maximum point.
Quadratic FunctionsLESSON 9-2
Additional Examples
ALGEBRA 1
(continued)
Step 2: Find the h-coordinate of the vertex.
The maximum height of the star will be about 731 ft.
Step 1: Find the x-coordinate of the vertex.
b2a– =
–882(–16) = 2.75
After 2.75 seconds, the star will be at its greatest height.
h = –16(2.75)2 + 88(2.75) + 610 Substitute 2.75 for t.
h = 731 Simplify using a calculator.
Quadratic FunctionsLESSON 9-2
Additional Examples
ALGEBRA 1
“Quadratic Functions” (9-2)
How do you graph a quadratic inequality.
Graph a quadratic inequality is similar to graphing a linear inequality. The parabola becomes the boundary line separating solutions from non-solutions. Just like a linear inequality, the curve (boundary line) is dashed if the inequality involves or and solid if the inequality involves ≥ or ≤. To figure out which side of the curve is shaded, test a point on each side to see if it is a solution of the inequality (in other words, makes the inequality a true statement).
Example: Graph y ≤ x2 - 3x – 4
Step 1: Graph the inequality. Use a solid line, because the inequality include the boundary line with ≤ (less than or equal)
Step 2: Check points on each side of the curve. Shade in the side in which the tested point makes the inequality true..
Test a point inside the curve, like (0, 0)
y ≤ x2 - 3x – 4 ; 0 ≤ 02 – 3(0) - 4
0 ≤ - 4 (not a true statement)
Test a point outside the curve, like (5, 0)
y ≤ x2 - 3x – 4 ; 0 ≤ 52 – 3(5) - 4
0 ≤ 25 – 15 – 4; 0 ≤ 6
0 ≤ 6 (true statement)
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ALGEBRA 1
Graph the boundary curve, y = –x2 + 6x – 5.
Use a dashed line because the solution of the inequality y > –x2 + 6x – 5 does not include the boundary.
Graph the quadratic inequality y > –x2 + 6x – 5.
Shade above the curve.
Quadratic FunctionsLESSON 9-2
Additional Examples
ALGEBRA 1
Graph each relation. Label the axis of symmetry and the vertex. Findthe domain and range.
2. ƒ(x) = –x2 + 4x – 21. y = x2 – 8x + 15
domain: set of all real numbers; range: {y : y ≥ –1}
domain: set of all real numbers; range: {y : y ≤2}
Quadratic FunctionsLESSON 9-2
Lesson Quiz
ALGEBRA 1
14
<3. y – x2 – 2x – 6
x = –4
domain: set of all real numbers; range: {y : y ≤ – 2}
Quadratic FunctionsLESSON 9-2
Lesson Quiz
