Modules 5 and 6: Linear Functions, Part I

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EXPLORE:

•m is the slope•b is the y-intercept

•Useful to find the x-intercept and the y-intercept

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Understanding the Different Forms of a Linear Function

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1. 2. 3.

Explain 1:

Example 1:

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4. 5. 6.

Example 2:

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Example 3:

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7. 8.

9. 10.

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Example 4:

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11. 12.

13. 14.

Modules 5 and 6: Linear Functions, Part II

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Explain 1:

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Example 1:

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Example 2:

Explain 2:

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Modules 5 and 6: Linear Functions, Part III

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(0, 4)

(3, 0)

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x-intercept: ___________

y-intercept: ________

x-intercept: ___________

y-intercept: ________

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x-intercept: ___________

y-intercept: ________

x-intercept: ___________

y-intercept: ________

Using the x and y-intercepts

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Write the equation of each line in standard form.

Explain 1:

Example 1:

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Write the equation of each line in standard form.

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•Useful to find the x-intercept and the y-intercept

Example 2:

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Example 3:

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Modules 5 and 6: Linear Functions in the Real World Context

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 =𝑓 𝑥2 − 𝑓 𝑥1

𝑥2 − 𝑥1

Another way to write it:

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Calculating Average Rate of ChangeExample 1

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1.

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2.

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3.

4.

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Example 2

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5.

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Example 3:

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7. 8.

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9.

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Example 4: Determine whether each function can be described by a linear function or not. If it is a linear function, then write the equation.

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Determine whether each function can be described by a linear function or not. If it is a linear function, then write the equation.

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11.

12.

13.

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14.

15.

16.

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Example 5

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17. 18.

19.

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Compare the initial value, the final value, the range, and the average range of change for each of the linear functions f(x) and g(x).

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Compare the initial value, the final value, the range, and the average range of change for each of the linear functions f(x) and g(x).

1.

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2.

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3.

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4.

5. 6.

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Linear Functions Absolute Value Functions

Quadratic Functions (parabolas)

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Father of all Linear Functions𝒇 𝒙 = 𝒙

Father of all Absolute Value functions

𝒇 𝒙 = 𝒙

Father of all Parabolas𝒇 𝒙 = 𝒙𝟐

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Transformation Rule (h and k are the parameters)

Applying translations to a function DO NOT change the SHAPE of the function, only its LOCATION .

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x f(x)

A -1 -1

B 0 0

C 1 1

x g(x)

A’ -1 1

B’ 0 2

C’ 1 3

2. Plot the following points on the coordinate plane.

Do they lie on the graph of f(x) or g(x)?

1. Using the concepts of slope and an y-intercept, find the equations of f(x) (green line) and g(x) (red line).

f(x) = _______ g(x) = __________

4. How can you obtain the points A’, B’, and C’ from A, B, and C?

3. Plot the following points on the coordinate plane. Do they lie on the graph of f(x) or g(x)?

5. Find h =_____ and k = ______.

6. Write the transformation rule: g(x) = _______________.

Explain 1:

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x f(x)

A -1 -1

B 0 0

C 1 1

x g(x)

A’ -3 -1

B’ -2 0

C’ -1 1

2. Plot the following points on the coordinate plane.

Do they lie on the graph of f(x) or g(x)?

1. Using the concepts of slope and an y-intercept, find the equations of f(x) (green line) and g(x) (red line).

f(x) = _______ g(x) = __________

4. How can you obtain the points A’, B’, and C’ from A, B, and C?

3. Plot the following points on the coordinate plane. Do they lie on the graph of f(x) or g(x)?

5. Find h =_____ and k = ______.

6. Write the transformation rule: g(x) = _______________.

Conclusion: Could a vertical translation f(x)+k be equivalent to the horizontal translation f(x-h) of the same linear function?

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1. Find the coordinates of the vertices of the functions f(x) (green) and g(x) (red).2. Compare the x and y coordinates of the vertices. Find h and k.3. Describe the transformation(s).4. Write the transformation rule.

h = ______ k = _______

f: V (__,__) g: V’(__,__)

Transformation: ____________________________________________

Rule: _________________

h = ______ k = _______

f: V (__,__) g: V’(__,__)

Transformation: ____________________________________________

Rule: _________________

h = ______ k = _______

f: V (__,__) g: V’(__,__)

Transformation(s): ____________________________________________

Rule: _________________

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1. Find the coordinates of the vertices of the functions f(x) (green) and g(x) (red).2. Compare the x and y coordinates of the vertices. Find h and k.3. Describe the transformation(s).4. Write the transformation rule.

h = ______ k = _______

f: V (__,__) g: V’(__,__)

Transformation: ____________________________________________

Rule: _________________

h = ______ k = _______

f: V (__,__) g: V’(__,__)

Transformation: ____________________________________________

Rule: _________________

h = ______ k = _______

f: V (__,__) g: V’(__,__)

Transformation(s): ____________________________________________

Rule: _________________

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Circle the word or expression that makes the statement true.

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4.

5.

6.

7.

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Compare the pair of functions given. Explain what transformations were done to f(x) to

obtain g(x). Determine h and k and write the rule g(x) = f(x-h) + k for each.

8.

10.

12.

9.

11.

13.

h = ______ k = _______

Transformation(s):________________ _______________________________

Rule: _________________

h = ______ k = _______

Rule: _________________

Transformation(s):________________ _______________________________

h = ______ k = _______

Transformation(s):________________ _______________________________

Rule: _________________

h = ______ k = _______

Rule: _________________

Transformation(s):________________ _______________________________

h = ______ k = _______

Transformation(s):________________ _______________________________

Rule: _________________

h = ______ k = _______

Rule: _________________

Transformation(s):________________ _______________________________