Super Duper Study Guide
VOCABULARY!
1. Axis of symmetry = the imaginary line that splits the graph of a quadratic function in half.
The equation is x = -b/2a from standard form, the equation is x = h in vertex form, and the
equation is x = (p+q)/2 in intercept form.
2. Completing the square = a process of solving a quadratic equation or of rewriting a
quadratic function from f(x) = ax2+bx+c to f(x) = a(x - h)2 + k.
3. Factoring = a process of solving quadratic equation or rewriting a quadratic function
from f(x) = ax2+bx+c to f(x) = a(x – p)(x - q).
4. Intercept form = f(x) = a(x – p)(x - q) where the points (p,0) and (q,0) are x-intercepts.
5. Maximum = the highest point of a graph.
6. Maximum value = the output value (y-value) of the maximum point.
7. Minimum = the lowest point of a graph.
8. Minimum value = the output value (y-value) of the minimum point.
9. Parabola = the symmetrical, u-shaped graphical representation of a quadratic function.
10. Quadratic function = a function in which the highest exponent of the input is 2/squared.
11. Quadratic formula = a formula for solving an equation ax2+bx+c=0; (-b ± √(b2 - 4ac))/2a
12. Root = another word for x-intercept of a quadratic/polynomial function or solution to
quadratic equation.
13. Solution = a value of x that makes the equation true.
14. Standard form = f(x) = ax2+bx+c where the vertex is ( -b/2a, f(-b/2a) ).
15. Vertex = the middle point of a parabola; maximum when opening downward and
minimum when opening upward.
16. Vertex form = f(x) = a(x - h)2 + k where the point (h,k) is the vertex.
17. X-intercept = where the graph crosses the x-axis (Y IS ZERO HERE!)
18. Y-intercept = where the graph crosses the y-axis (X IS ZERO HERE!)
19. Zero = another word for x-intercept of a quadratic/polynomial function or solution to
quadratic equation.
Find the axis of symmetry and vertex of the following functions. State whether the vertex is a
maximum or a minimum. State what form the function is in.
1. 𝐹(𝑥) = (𝑥 − 1)2
2. 𝐺(𝑥) = −(𝑥 + 2)2 − 4
3. 𝐻(𝑥) = 𝑥 2 + 12𝑥 − 1
Find the roots/zeros/x-intercepts of the following functions. State what form the function is in.
1. 𝐽(𝑥) = −𝑥 2 + 12𝑥 + 13
2. 𝐾(𝑥) = 2(𝑥 − 5)(𝑥 + 4)
3. 𝐿(𝑥) = −(𝑥 − 3)2 + 16
PRACTICE!
Solve the following using the given method.
1.
2.
3.
4.
5.
2𝑥 2 − 10𝑥 − 48 = 0 by factoring
𝑥 2 − 6𝑥 − 112 = 0 by completing the square
3𝑥 2 + 15𝑥 − 42 = 0 by using the quadratic formula
2𝑥 2 + 20 = 𝑥 2 + 120 by taking the square root
3𝑥 2 + 7𝑥 + 4 = 0 by any method
Graph the following functions.
6. 𝐹(𝑥) = (𝑥 + 2)(𝑥 − 5)
7. 𝐺(𝑥) = (𝑥 + 5)2 + 4
1
8. 𝐻(𝑥) = − 2 (𝑥 + 3)(𝑥 + 5)
9. 𝐽(𝑥) = −3𝑥 2 − 6𝑥 + 24
10. 𝐾(𝑥) = −2(𝑥 − 1)2 + 3
Answer the following questions about falling objects and maximization/
minimization.
11. An object is launched at 19.6 meters per second (m/s) from a 58.8-meter
tall platform. The equation for the object's height s at time t seconds after
launch is s(t) = –4.9t2 + 19.6t + 58.8, where s is in meters. When does the
object strike the ground?
12. An object in launched directly upward at 64 feet per second (ft/s) from a
platform 80 feet high. What will be the object's maximum height? When
will it attain this height?
13. The number of bacteria in a refrigerated food is given by N(T) = 20T2 – 20T
+ 120, for − 2 ≤ T ≤ 14 and where T is the temperature of the food in Celsius.
At what temperature will the number of bacteria be minimal?
14. A manufacturer of tennis balls has a daily cost of C(x) = 200 −10x + 0.01x2
where C is the total cost in dollars and x is the number of tennis balls
produced. What number of tennis balls will produce the minimum? What is
the minimum cost?
15. A foul ball leaves the end of a baseball bat and travels according to the
formula h(t) = 64t – 16t2 where h is the height of the ball in feet and t is the
time in seconds. How long will it take for the ball to reach a height of 64
feet in the air?