10-2 Quadratic Functions NOTES
In lesson 10-1, you investigated the graphs of 𝑦 = 𝑎𝑥 2 and 𝑦 = 𝑎𝑥 2 + 𝑐. In the quadratic
function 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, the value of 𝑏 affects the position of the axis of symmetry.
Consider the graphs of the following functions.
𝑦 = 2𝑥 2 + 2𝑥
𝑦 = 2𝑥 2 + 4𝑥
𝑦 = 2𝑥 2 + 6𝑥
Notice that all three graphs have the same y-intercept. This is because in all three equations
𝑐 = 0. The axis of symmetry changes with each change in the 𝑏 value. The equation of the
𝑏
axis of symmetry is related to the ratio 𝑎.
Equation:
𝑦 = 2𝑥 2 + 2𝑥
𝑦 = 2𝑥 2 + 4𝑥
𝑏
:
𝑎
Axis of symmetry:
1 𝑏
−𝑏
2 𝑎
2𝑎
The equation of the axis of symmetry is 𝑥 = − ( ) or
𝑦 = 2𝑥 2 + 6𝑥
10-2 Quadratic Functions NOTES
Graph of a Quadratic Function
The graph of 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, where 𝑎 ≠ 0, has the line 𝑥 =
symmetry. The x-coordinate of the vertex is
−𝑏
2𝑎
−𝑏
2𝑎
as its axis of
.
When you substitute 𝑥 = 0 into the equation 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, 𝑦 = 𝑐. So the y-intercept of a
quadratic function is the value of c. You can use the axis of symmetry and the y-intercept to
help you graph a quadratic function.
Example 1: Graphing 𝒚 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
1.) Graph the function 𝑦 = −3𝑥 2 + 6𝑥 + 5
Step 1: Find the equation of the axis of symmetry:
Step 2: Find the coordinates of the vertex:
Step 3: Find two other points.
a.) Use the y-intercept
b.) Choose a value for x on the same side of the vertex as the y-intercept
Step 4: Reflect your two points across the axis of symmetry to get two more points.
Step 5: Draw the parabola
10-2 Quadratic Functions NOTES
Graph the function. Label the axis of symmetry and the vertex.
2.) 𝑦 = 𝑥 2 − 6𝑥 + 9
3.) 𝑦 = 2𝑥 2 + 4𝑥 − 3
4.) 𝑦 = −𝑥 2 + 4𝑥 − 2
10-2 Quadratic Functions NOTES
You saw in the previous lesson that the formula ℎ = −16𝑡 2 + 𝑐 describes the height above the ground
of an object falling from an initial height 𝑐, at time 𝑡. If an object is given an initial upward velocity 𝑣
and continues with no additional force of its own, the formula ℎ = −16𝑡 2 + 𝑣𝑡 + 𝑐 describes its
approximate height above the ground.
Example 2: Real-World Problem Solving
1.) Fireworks: In professional fireworks displays, aerial fireworks carry “stars” upward, ignite them, and
project them into the air. Suppose a particular star is projected from an aerial firework at a starting
height of 520 feet with an initial upward velocity of 72 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 = −16𝑡 2 + 72𝑡 + 520 gives the star’s height ℎ in feet at time 𝑡 in seconds. Since the
coefficient of 𝑡 2 is negative, the curve opens downward, and the vertex is the maximum point.
Step 1: Find the t-coordinate of the vertex.
Step 2: Find the h-coordinate of the vertex.
2.) A ball is thrown into the air with an initial upward velocity of 48 ft/s. Its height ℎ in feet after 𝑡
seconds is given by the function ℎ = −16𝑡 2 + 48𝑡 + 4.
a.) In how many seconds will the ball reach its
maximum height?
b.) What is the ball’s maximum height?
3.) 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 the star to reach its maximum height? How far
above the ground will it be?