Lesson 3.2: Investigating Quadratic Functions in Standard Form
Specific Outcome: Analyze quadratic functions of the form y = ax2 + bx + c to identify
characteristics of the corresponding graph, including: the vertex, domain and range,
direction of opening, axis of symmetry, x- and y- intercepts and to solve problems.
(Achievement indicators ~ 4.1 – 4.8)
A quadratic function of the form f(x) = ax2 + bx + c is written in STANDARD FORM.
PART A:
Using your calculator, graph the functions that result from substituting the following avalues into the quadratic function, f(x) = ax2 + 4x + 5
a = -4
a=1
PART B:
Using your calculator, graph the functions that result from substituting the following
c – values into the function f(x) = -x2 + 4x + c
c = 10
c=0
c = -5
The Standard form of a quadratic function is f(x) = ax2 + bx + c or y = ax2 + bx + c, where
a, b, and c are real numbers with a ≠ 0.
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a determines the shape and whether the graph opens upward (positive a) or
downward (negative a)
b influences the position of the graph
c determines the y-intercept of the graph
Example 1: For each graph of a quadratic function identify the following:
f ( x)   x 2  2 x  8
a) The direction of opening: _________________________
b) The coordinates of the vertex: _____________________
c) The maximum or minimum value: __________________
d) The equation of the axis of symmetry: _______________
e) The x-intercepts and y-intercepts: __________________
f) The domain and range: __________________________
Example 2: For each graph of a quadratic function identify the following:
f ( x )  2 x 2  12 x  25
a. The direction of opening: ______________________
b. The coordinates of the vertex: __________________
c. The maximum or minimum value: ________________
d. The equation of the axis of symmetry: ____________
e. The x-intercepts and y-intercepts: _______________
f. The domain and range: ________________________
Example 3: A frog sitting on a rock jumps into a pond. The height, h, in centimeters, of
the frog above the surface of the water as a function of time, t, in seconds, since it
jumped can be modeled by the function h(t) = -490t2 + 150t + 25. Where appropriate,
answer the following questions to the nearest tenth.
a) Graph the function on your graphing calculator.
b) What is the y-intercept? What does it represent in this situation?
c) What maximum height does the frog reach? When does it reach that height?
d) When does the frog hit the surface of the water?
e) How high is the frog 0.25 seconds after it jumps?
Example 4: A farmer has 60 metres of fencing available to make a corral along the side
of his barn. What is the maximum area that can be enclosed if one side of the corral is
the barn wall?
Practice Questions: Page 174 # 2, 5(a), 7, 12, 15