Big Ideas Examples

Big Ideas Examples Solving Equations: Compose Both Sides with Inverse Functions in Reverse Order Linear 5x – 3 = 2x + 9

First, put the equation in a form where we can use this

3x – 3 = 9

approach. To evaluate the left side we would use the

3x

functions 3x and x – 3 in that order. We use the inverse

 3  3  9  3

3x = 12

3x  3



function x + 3 first, then the inverse function

12 3

x . 3

Note that there is “cancellation” at each step.

x=4 Quadratic  2x  3  5  7 2

Evaluate: 1) x  3 , 2) x 2 , 3)  2 x , 4) x  5 Solve: 1) x  5 , 2)

x , 3) 2

x , 4) x  3

Radical

5 2x  3  1  9

Evaluate: 1) 2x, 2) x – 3, 3) Solve: 1) x  1 , 2)

x , 4) 5 x , 5) x  1

x x , 3) x 2 , 4) x  3 , 5) 5 2

Exponential

4e x / 2  3  5  7

Evaluate: 1)

x , 2) x  3 , 3) e x , 4) 4 x , 5) x  5 2

Solve: 1) x  5 , 2)

x , 3) ln(x) , 4) x  3 , 5) 2 x 4

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Functions as Mappings I.

Definitions A. A relation is a mapping f : X  Y that maps elements of the set X to elements B. C. D. E.

of the set Y. A function is a relation where no element x in X is mapped to two elements in Y. The domain of a relation is the set of elements of X mapped to an element of Y. The range of a relation is set of elements of Y that are mapped to by an element of the domain of the relation. The implied domain of a function f : X  Y with mapping rule x  f (x) is

x X |

II.

f ( x)  Y . Consequences of Using This Definition A. Definition of a discrete function:

B. Addition of real numbers is a function. Why?

C. What happens if X (Y) is not a set of numbers?

D. Is differentiation of functions a function?

E. How would you define the graph of a function?

F. What are other consequences?

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Modeling Converting Graphs to Tables to Equations: Translation of Axes Linear

y' 6

y = f(x) 4

2

10

5

-5

-10

x'

-2

-4

-6

x 2 3 4

x2 0 1 2

y 1 3 5

Equation: y  1   2x  2

y 1 0 2 4

(Point-Slope Form)

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Quadratic 8

y' 6

x' 4

2

y = f(x)

-10

-5

5

10

-2

-4

x 1 1 0 1

y

x 0 1 2

3 5 3

Equation: y  5   2x  1

2

y5 2 0 2

(Vertex Form)

4

Square Root 6

4

y' 2

y = f(x) -10

-5

5

10

-2

-4

x' -6

x 3 2

y

4 2

x3 0 1

y4 0 2

Equation: y  4  2 x  3

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Absolute Value

4

y'

2

-10

-5

5

10

y = f(x) -2

x' -4

-6

x 2 3 4

y

3  2.5 2

x2 0 1 2

y3 0 0.5 1

Equation: y  3  0.5x  2

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Reciprocal 8

y' 6

y = f(x) 4

2

x' -10

-5

5

10

-2

-4

-6

-8

-10

y

x 5 4 2

Equation: y  2 

2 5 3

(center)

x5 0 1 3

y2 0 3 1

3 x5

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Other Transformations Reflection Across An Axis 8

6

(x, y)

(-x, y) 4

2

-10

-5

5

10

-2

-4

(x, -y) -6

-8

-10

Reflection of y  f (x) Across x-Axis:  y  f (x) Reflection of y  f (x) Across y-Axis: y  f ( x) Example: y  2 x 2  3x  4

8

Stretch Away From x-Axis; Compression Toward x-Axis

8

Stretch

(x, 2y)

6

(x, y)

4

2

-10

-5

Compression

5

-2

Reflection & Compression

(x, 0.5y)

10

(x, -0.5y)

-4

-6

-8

Reflection & Stretch

(x, -2y)

Stretch of y  f (x) away from x-axis: y  a f (x) where a  1 Compression of y  f (x) toward x-axis: y  a f (x) where a  1

  Example: y  3   2 sin x   6 

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Problem Solving Polya’s four-step heuristic works in any context 1. 2. 3. 4.

Understand the problem Devise a plan Carry out the plan Look back

Absolute Value as Distance From the Origin 1. 1-space: x 

x2

2. 2-space: x, y   x 2  y 2 3. 3-space: x, y, z   x 2  y 2  z 2 4. Distance between two points: Translate one point to origin. Now it is an absolute value problem. Space does not matter; process is the same. 5. “Circle”: Locus of points equidistant from a fixed point called the center. 6. 1-space circle with center at origin: x  x 2  r or x 2  r 2 7. 1-space circle with center at h: Translate center to origin; x  h 

x  h2

 r or

x  h2  r 2

8. 2-space circle with center at h, k  :

9. 3-space circle with center at h, k , l  :

10. How would you define disks in 1-space, 2-space, 3-space?

11. What is the connection to vectors in different spaces?

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