Download Exam 2 Review 2.1-2.5, 3.1-3.4 2.1:Coordinate Geometry 2.2: Linear

Exam 2 Review
2.1-2.5, 3.1-3.4
2.1:Coordinate Geometry
• Memorize 2 formulas for two points (x1 , y1 ) and (x2 , y2 ):
Midpoint formula:
x1 + x2 y1 + y2
,
2
2
p
(x2 − x1 )2 + (y2 − y1 )2
Distance formula:
d=
1. Find the midpoint and distance between (−3, 4) and (6, −2).
2. Find the midpoint and distance between
7
2 , −5
and (3, −4).
2.2: Linear Equations and Inequalities
• Be able to find x and y intercepts and be able to describe in words what x and y intercepts are.
3. Graph 6x + 8y ≤ −12.
4. Graph 4x − 3y < 16.
1
2.3: Determining the Equation of a Line
2
• Memorize characteristics of 3 types of equations of lines:
– Point-Slope Form: (y − y1 ) = m(x − x1 )
◦ m is the slope
◦ x and y are the most general x and y
◦ (x1 , y1 ) is any specific point on the line
– Slope-Intercept Form: y = mx + b
◦ m is the slope
◦ x and y are the most general x and y
◦ b is the y-intercept
– Standard Form: Ax + By = C
◦ A, B, and C are all integers, A ≥ 0
◦ x and y are the most general x and y
• Memorize Property 2.1
– Two lines with slopes m1 and m2 are:
◦ parallel if m1 = m2
◦ perpendicular if m1 m2 = −1
5. Find the equation of the line determined by the points (3, 3) and (5, −7). Put your answer in slopeintercept form.
6. Find the equation of the line which has y intercept −2 and slope 3/4. Put your answer in standard
form.
7. Find the equation of the line parallel to 4x − 5y = 15 that goes through the point (6, 13). Put your
answer in point-slope form.
8. Find the equation of the line perpendicular to y = − 85 x − 1 through the point (2, 1). Put your answer
in standard form.
2.4: Graphing Techniques
3
• Memorize rules for determining symmetry
– A graph has y-axis symmetry if replacing x with -x results in an equivalent equation
– A graph has x-axis symmetry if replacing y with -y results in an equivalent equation
– A graph has origin symmetry if replacing both x with -x and y with -y results in an equivalent
equation
• Be familiar with common restrictions:
– Avoid
√
anything
and negative
0
• Look at resulting values in common graphs:
√
– ◦ positive ≥ 0
– ◦ (anything)2 ≥ 0
– ◦ |anything| ≥ 0
9. Graph y = −2|x − 3| + 1.
10. Graph y =
11. Graph y =
√
x2 + x − 2.
x+1
.
x2 + x − 6
4
2.5: Circles
• Memorize the standard form for the equation of a circle with center at (h, k) and radius r.
(x − h)2 + (y − k)2 = r2
• No ellipses or hyperbolas will be on the exam
• Complete the square with two variables
12. Write the equation for this circle in standard form. Give the center and radius then graph the circle.
2x2 − 16x + 2y 2 + 20y + 58 = 0
13. Write the equation for this circle in standard form. Give the center and radius then graph the circle.
1 2
1
x − 6x + y 2 + 2y + 36 = 0
4
4
14. Write the equation for this circle in standard form. Give the center and radius then graph the circle.
x2 − x + y 2 + 3y −
11
=0
2
3.1: Functions
5
• Domain is the set of all possible inputs x
• Range is the set of all possible outputs f (x)
• A function is even if it has y-axis symmetry. That is, a function is even if f (−x) = f (x)
• A function is odd if it has origin symmetry. That is, a function is odd if f (−x) = −f (x).
15. Give the domain and range of the following functions. Is each function odd, even, or neither?
√
◦ f (x) = x − 2
◦ g(x) = (x − 1)3
◦ h(x) =
√
x2 + 1 − 4
3.2: Linear Functions
• Be comfortable graphing a line in slope-intercept form
• Be able to explain what f (x) means and what f (2) means
16. Graph f (x) = − 14 x + 7. What is f (4)?
17. Stephen makes $500 each month plus $50 for every cell phone he sells. Write a linear function that
describes his situation. How much money does he make if he sells 4 cell phones? What is the domain?
3.3: Quadratic Functions
6
• Be able to explain what each part of vertex form y = a(x − h)2 + k means
◦ y = x2 + k is a vertical shift
◦ y = ax2 is a stretch, scrunch, or flip depending on the value of a
◦ y = (x − h)2 is a horizontal shift
• Be able to graph a parabola in vertex form
• Complete the square to write f (x) = ax2 + bx + c in vertex form
18. Graph f (x) = −(x − 5)2 + 3
19. The basic graph of f (x) = x2 is scrunched (made narrower) by a factor of 2, is moved 6 units down,
and 3 units left. Write the new quadratic function this new graph illustrates.
20. Put f (x) = 3x2 + 6x − 1 in vertex form then graph the function.
3.4: More Quadratic Functions
7
• Be able to graph a parabola using the techniques from this chapter
• Memorize the formula for the vertex of the parabola f (x) = ax2 + bx + c
b 4ac − b2
− ,
2a
4a
• Be able to find the minimum or maximum value of a function based on a story problem
21. What is the vertex of f (x) = −2x2 − 4x + 2? Does this vertex represent a minimum or a maximum?
22. You have 1200 feet of fencing available and you want to make a rectangular pen which is divided into 5
smaller pens of equal size. The fencing used to divide the larger area into smaller pens must be parallel
to 2 sides of the rectangle. Write a quadratic equation that illustrates this situation. What are the
dimensions of the rectangular pen that will maximize the area?