Curve Sketching Review

Review for Final Exam Part 2

Hey, how are ya?

Questions on the Exam

1) Graphing Equations
2) End behavior 
3) Real World Problems
4) Transformations of Graph
5) Write equation of a hyperbola.
6) Write equation of an ellipse.
7) Conic Section type from Diagram
8) Rational Functions Behavior Around Asymptotes x2
9) Solve Rational Inequalities

Questions on the Exam

10) Properties of Logarithms

11) Solve Exponentials

12) Converting to & from Parametric Equations

13) Graph Parametric Equations

14) Composite Functions

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15) Polar Equations

16) Arithmetic and Geometric series in recursive notation.

17) Arithmetic and Geometric series in sigma notation.

18) nth Term of a Geometric series

19) Evaluate Finite sums using sigma notation.

Conic Section Diagrams

Ellipse Equation

 $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$  
Vertical:
 $b>a$  
Horizontal:
 $a>b$  
Focus Distance:
 $c^2=a^2-b^2$  

Multiple Choice

Determine the Foci for the ellipse.

• (-3 , 4) and (-3 , 0)

• (3 , -4) and (3 , 0)

• (-3 , -4) and (-3 , 0)

• (-3 , 4) and (3 , 0)

Multiple Choice

What equation helps us find the focus points on an ELLIPSE?

• c² = a² + b²

• c² = a² - b²

• 1/4d

• (x-h)² + (y-k)² = r²

Multiple Choice

Which of the following is the graph of  $$\frac{\left(x+3\right)^2}{4}+\frac{\left(y+1\right)^2}{9}=1$$  

Multiple Choice

Find the standard equation of the given ellipse

Multiple Choice

What is the center of the ellipse?

• (0, 0)

• (1, 5)

• (1, 0)

• (0, 1)

Solve Exponentials

1) Get the power by itself by solving for it.

2) Use the inverse logarithm of the base to get the exponent by itself.

3) Solve the resulting equation.

Multiple Choice

Solve for b:
36^3b = 216^b+4

• 6
• 1
• 4
• -4

Multiple Choice

Solve for x

• 3

• 4

• 9

• 27

Multiple Choice

Solve for x:

2^{3x - 1} = 32

• 0

• -1

• 5/3

• 2

Properties of Logarithms

 $\log_b\left(b\right)=1$  
 $\log\left(mn\right)=\log\left(m\right)+\log\left(n\right)$  
 $\log\left(\frac{m}{n}\right)=\log\left(m\right)-\log\left(n\right)$  
 $\log\left(m^n\right)=n\log\left(m\right)$  
 $\log_b\left(m\right)=\frac{\log_c\left(m\right)}{\log_c\left(b\right)}$  or  $\log_b\left(m\right)\cdot\log_c\left(b\right)=\log_c\left(m\right)$  

Multiple Choice

Write the logarithm expression as a single logarithm

log₄60 - log₄4 + log₄x

• log₄15

• log₄15x

• log₄56x

• xlog₄30

Multiple Choice

Simplifying using the laws of logs

• log(48)

• log(4)

• log(80)

• log(8)

Multiple Choice

Simplifying using the laws of logarithms

• log(10)

• log(21)

• log(343)

• log(4)

Parametric Graphs

We need the path and the direction a point would follow on that path with the given equations.

Multiple Choice

Sketch the curve of x(t) = 2t + 1 and y(t) = t² + 2.

• A

• B

• C

• D

Multiple Choice

Eliminate the parameter.

x= 2+4t and y=-1+6t

• y=(3/2)x - 4

• t=(x-2)/4

• y=x - 4

• y= (2/3)x + 4

Multiple Choice

Write x = 2t  and y = t² + 3  in rectangular form .

• y = 4x²  + 3
• y = ¼ x² + 3
• y = 4t² + 12
• y = (x - 2)² + 3

Multiple Choice

Write x = 2t  and y = t² + 3  in rectangular form .

• y = 4x²  + 3
• y = ¼ x² + 3
• y = 4t² + 12
• y = (x - 2)² + 3

Composite Functions

 $f\left(x\right)=3x$  &  $g\left(x\right)=2x^2-3$  
 $f\ \circ\ g\left(x\right)=f\left(g\left(x\right)\right)=3\left(2x^2-3\right)=6x^2-9$  
 $g\ \circ f\left(x\right)=g\left(f\left(x\right)\right)=2\left(3x\right)^2-3=18x^2-3$  

Multiple Choice

Given f(x) = 2x and g(x) = x²+3 , find f(g(x)).

• x²+2x+3
• 4x²+3
• 2x²+3
• 2x²+6

Multiple Choice

If f(x) = x-1 and g(x) = 2x,

what is and g(f(x)) ?

• 2x²-1

• 2x-1

• 2x-2

• x-2

Multiple Choice

• 9 - √17
• 4
• 2
• √8

Multiple Choice

Determine which two functions are composed so that f(g(x)) = h(x), where h(x) = (x-3)² - 5.

• f(x) = x - 3 g(x) = x² - 5

• f(x) = (x - 3)² g(x) = x - 5

• f(x) = x² - 5 g(x) = (x - 3)

• f(x) = x² g(x) = (x - 3) - 5