Midterm Study Guide

Equation of a Circle

(x - h)^2 + (y - k)^2 = r^2

Standard form for equation of a circle

(x - h)^2 + (y - k)^2 = r^2

Quadratic Formula

-b±[√b²-4ac] / 2a

b^2-4ac = 0

1 real solution Reminder 7x^2=-6x-6 a = 7 b = -6 c = -6

Steps to find the standard form for equation of a circle

1. Find the point in the center of the circle 2. Fill in the equation with the point (x - h)^2 + (y - k)^2 = r^2 REMINDER: x is h, y is k. 3. Distribute any signs Example. (-4, 2) r = 3 Solution (x + 4)^2 + (y - 2)^2 = 3^2

How can the graph of f(x) = -(x - 8)^2 + 2 be obtained from the graph of y = x^2?

1. Reflect over the x - axis 2. Shift 8 units to the right and 2 units up REMINDER Shrink/Stretch, Reflect, Shift

What are the steps to solve: |2x + 6| < 18

1. Rewrite the problem for two solutions |2x + 6| < 18 and -(2x + 6) < 18 2. Solve for both solutions REMINDER: Distribute the negative REMINDER: Flip sign if dividing by negative 3. Plug in both solutions as x to verify they are solutions (-12,6)

What are the steps to solve: |5x - 2| ≥ 9

1. Rewrite the problem for two solutions |5x - 2| ≥ 9 and -(5x - 2) ≥ 9 2. Solve for both solutions REMINDER: Distribute the negative REMINDER: Flip sign if dividing by negative 3. Plug in both solutions as x to verify they are solutions REMINDER: ≥ means there will be a Union (U) (-∞, -7/5]U[11/5, ∞) or (-∞, -1.4]U[2.2, ∞)

What are the steps to solve: |x + 6| - 3 = 14

1. Rewrite the problem for two solutions |x + 6| - 3 = 14 and -(x + 6) - 3 = 14 2. Solve for both solutions REMINDER: Distribute the negative 3. Plug in both solutions as x to verify they are solutions -23, 11

How can the graph of f(x) = 0.1(x + 3)^2 - 10 be obtained from the graph of y = x^2?

1. Shrink vertically by a factor of 0.1 2. Shift 3 units left and down 10 units REMINDER Shrink/Stretch, Reflect, Shift

Steps for problems such as: -20 < 5x + 5 ≤ 0

1. Subtract ( 5 ) from each section 2. Divide ( 5 ) from each section so the x is alone REMINDER change sign when dividing by negatives 3. Graph the solution set REMINDER pay attention to using ( ) and / or [ ]

b^2-4ac < 0

2 imaginary solutions Reminder 7x^2=-6x-6 a = 7 b = -6 c = -6

b^2-4ac > 0

2 real solutions Reminder 7x^2=-6x-6 a = 7 b = -6 c = -6

Relative maxima

A point on the graph of a function where no other nearby points have a greater y-coordinate. Written as example: (0, 1) relative maxima 1 at x = 0

Relative minima

A point on the graph of a function where no other nearby points have a lesser y-coordinate. Written as example: (2,-3) relative minima -3 at x = 2

Presented Question: A car rental company charges $29 per day to rent a particular type of car and $0.14 per mile. Juan is charged $46.50 for a one-day rental. How many miles did he drive? How would you solve this?

Create a formula 46.50 = 0.14x + 29 Solve for solution 46.50 - 29 = 0.14x 17.50 = 0.14x 17.50/0.14 = 0.14x/0.14 125 = x 125 miles

Presented Question: A projectile is thrown upward so that its distance above the ground after t seconds is h(t) = -16t^2 + 476t. After how many seconds does it reach its maximum height? How would you solve this?

Notice that this is a parabola Use the vertex formula Round one decimal place REMINDER since this deals with time, answer cannot be negative t = -b/2a t = -476/2(-16) t = 119/8 t = 14.876 t = 15 seconds

Presented Question: The resistance of a wire varies directly as the length of the wire and inversely as the square of the diameter of the wire. A 20 ft. length of wire with a diameter of 0.1 inches has resistance of 3 ohms. What would the resistance be for a 21 ft. length, with a diameter of 0.01 inch? How would you solve this?

R = Lk/d^2 (resistance = x * length / diameter squared) Plug in the information provided Multiply both sides by the inverse to clear fraction 3 = 20k / (0.1)^2 (0.1^2 / 20)3 = 20k / (0.1)^2(0.1^2 / 20) 3(0.1^2 / 20) 3(0.01 / 20) TUTOR

Presented Question: A rectangular city park has a jogging loop that goes along a length, width, and diagonal of the park. To the nearest yard, find the length of the jogging loop, if the length of the park is 125 yards and its width is 75 yards. How would you solve this?

See the length and width as points on a graph: (0, 125) (0, 75) Use the distance formula Solve Turn the radical into a decimal Round the decimal one place if needed Add the length and width to the answer REMINDER put the word yards 25√34 = 145.77 145.77 = 146 146 + 125 + 75 = 346 346 yards

Presented Question: The cost of manufacturing a molded part is related to the quantity produced during a production run. When 100 parts are produced, the cost is $300. When 400 parts are produced, the cost is $2700. What is the average cost per part? How would you solve this?

See the two portions as points on a graph: (100, 300) and (400, 2700) Use the slope formula Solve for solution REMINDER Don't forget to use the $ in the answer. 2700-300/400-100 = 2400/300 = $8

Order for Transformational Graphing

Stretch or Shrink Reflect Shift

Intervals of Decreasing

The interval or intervals when the function is decreasing across the x-values. (0,1) and (2,-3) Decreasing (0,2)

Intervals of Increasing

The interval or intervals when the function is increasing across the x-values. (0,1) and (2,-3) Increasing (-∞,0),(2,∞)

Presented Question: The weight that a horizontal beam can support varies inversely as the length of the beam. Suppose that a 2-m beam can support 630 kg. How many kilograms can a 1-m beam support? How would you solve this?

Use the inverse formula y = k/x Write the formula as W = k/L (W = weight, L = length) Substitute with the provided information Solve for k Plug information into formula for question solution 630 = k/2 2(630) = (k/2)2 1260 = K W = 1260/L W = 1260/1 W = 1260 1260 kg

Distance Formula

d = √[( x₂ - x₁)² + (y₂ - y₁)²]

Slope Formula

m = (y₂- y₁) / (x₂- x₁)

Vertex Formula

x = -b/2a

Inverse Formula

y= k/x

Direct Formula

y=kx