1.01b Parent Functions Worksheet.pdf - 1.01b Parent Function ... - Free Printable

Let’s go step by step through each part of the worksheet.

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Part 1: Identify the name of the parent function

We are given equations and asked to name the *parent function* — that means the simplest form of the function type, without any shifts, stretches, or reflections.

a) g(x) = x + 4
→ This is a linear function. Parent function: f(x) = x → Name: Linear

b) f(x) = -x²
→ This is a quadratic (has x²). The negative sign flips it, but parent is still f(x) = x² → Name: Quadratic

c) h(x) = √(x - 3)
→ Square root function. Parent: f(x) = √x → Name: Square Root

d) p(x) = |x| / 2
→ Absolute value divided by 2. Parent: f(x) = |x| → Name: Absolute Value

e) q(x) = 1/x
→ Reciprocal function. Parent: f(x) = 1/x → Name: Reciprocal (or Rational)

f) r(x) = x³ - 5
→ Cubic function. Parent: f(x) = x³ → Name: Cubic

g) s(x) = 2ˣ
→ Exponential with base 2. Parent: f(x) = 2ˣ → But usually we say the general parent is f(x) = bˣ, so name: Exponential

h) t(x) = log₂(x)
→ Logarithmic with base 2. Parent: f(x) = log₂(x) → Name: Logarithmic

i) u(x) = 3√x
→ Cube root? Wait — notation says “3√x” — if it’s written as 3 times square root of x, then it’s still square root family. But if it’s cube root, it would be written as ∛x. Looking at context, likely it’s cube root because otherwise it’s same as c). Let me check standard notation.

Actually, in many textbooks, “3√x” means cube root of x. So parent: f(x) = ∛x → Name: Cube Root

j) v(x) = 1/(x - 7)²
→ This is reciprocal squared. Parent: f(x) = 1/x² → Name: Reciprocal Squared (sometimes called "Inverse Square")

But let’s confirm common naming:

Standard parent functions:

- Linear: f(x) = x
- Quadratic: f(x) = x²
- Cubic: f(x) = x³
- Square Root: f(x) = √x
- Cube Root: f(x) = ∛x
- Absolute Value: f(x) = |x|
- Reciprocal: f(x) = 1/x
- Reciprocal Squared: f(x) = 1/x²
- Exponential: f(x) = bˣ (usually b=2 or e)
- Logarithmic: f(x) = log_b(x)

So for j), since it’s 1 over (x-7) squared, the parent is 1/x² → Reciprocal Squared

Now list them clearly:

a) Linear
b) Quadratic
c) Square Root
d) Absolute Value
e) Reciprocal
f) Cubic
g) Exponential
h) Logarithmic
i) Cube Root
j) Reciprocal Squared

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Part 2: Give an example of an equation not covered by the following family of functions

This means: pick a function type that is NOT one of the ones listed above (linear, quadratic, cubic, sqrt, abs val, reciprocal, exp, log, etc.)

Common missing families:

- Trigonometric: sin, cos, tan
- Piecewise
- Constant
- Step function
- Polynomial higher than cubic (like quartic)
- Radical other than square/cube root (like fourth root)

The question says: “Give an example... (there are many correct answers)”

So we can pick any.

Examples:

a) absolute value → already covered → pick something else → e.g., sin(x)

b) cubic → covered → pick cos(x)

c) logarithmic → covered → pick tan(x)

d) exponential → covered → pick constant function like f(x) = 5

Wait — actually, looking back, the instruction says:

“Give an example of an equation not covered by the following family of functions.”

And then lists:

a) absolute value
b) cubic
c) logarithmic
d) exponential

So for each letter, give an example of a function that is NOT in that family.

So:

a) Not absolute value → e.g., f(x) = x² (quadratic) — but wait, quadratic wasn’t mentioned yet? Actually, the “following family” refers to the four listed: abs val, cubic, log, exp.

So for each, pick a function that does NOT belong to that specific family.

But the way it’s phrased: “not covered by the following family of functions” — and then lists a,b,c,d — probably meaning: for each of these types, give an example of a function that is NOT that type.

But that doesn’t make sense — because every function is not some other type.

I think it means: Here are four function families (abs val, cubic, log, exp). For each, give an example of a function that belongs to a DIFFERENT family — i.e., not that one.

But that seems odd.

Alternative interpretation: Maybe it’s saying: “Here are four function families. Give one example of a function that is NOT in ANY of these four families.”

That makes more sense.

Because it says: “an equation not covered by the following family of functions” — singular “family”, but lists four. Probably typo, should be “families”.

And then says “(there are many correct answers)” — implying one answer total, not per letter.

Looking at layout:

It says:

Give an example of an equation not covered by the following family of functions. (there are many correct answers)

Then lists:

a) absolute value
b) cubic
c) logarithmic
d) exponential

Probably, it’s listing examples of families that ARE covered, and we need to give one function that is NOT any of those.

So avoid: absolute value, cubic, logarithmic, exponential.

Pick, say, trigonometric: f(x) = sin(x)

Or quadratic: f(x) = x² — but quadratic was in part 1, so maybe allowed? The instruction doesn’t say to avoid previously mentioned, just not these four.

To be safe, pick trigonometric.

Example: f(x) = sin(x)

Or even simpler: f(x) = 5 (constant function) — not in any of those four.

Constant is fine.

I’ll go with f(x) = sin(x) as it’s clearly different.

But let’s see what’s expected. Since it’s open-ended, any non-abs-val, non-cubic, non-log, non-exp is fine.

Another good one: f(x) = x⁴ (quartic polynomial)

I’ll choose f(x) = sin(x)

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Part 3: Name the parent function for each graph

Left graph: V-shaped, vertex at origin, opens up → classic absolute value → Absolute Value

Right graph: straight line through origin, slope positive → Linear

Wait — right graph has points: looks like passes through (0,0), (1,1), (-1,-1) → yes, y=x → linear.

Left graph: symmetric about y-axis, vertex at (0,0), goes through (1,1), (2,2)? No — for absolute value, at x=1, y=1; x=2,y=2 — but in graph, if it's V-shape with lines going to (1,1) and (-1,1), then yes, |x|.

In left graph, if it goes through (1,1) and (-1,1), and vertex at (0,0), then it’s |x|.

Sometimes graphs might be shifted, but here both seem centered at origin.

Left: Absolute Value
Right: Linear

But let’s double-check.

Left graph: two rays from origin, one with slope 1 for x>0, slope -1 for x<0 → definitely |x| → Absolute Value

Right graph: straight line through origin with constant slope → Linear

Yes.

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Now, compiling all answers.

First part: names of parent functions

a) Linear
b) Quadratic
c) Square Root
d) Absolute Value
e) Reciprocal
f) Cubic
g) Exponential
h) Logarithmic
i) Cube Root
j) Reciprocal Squared

Second part: example not covered → I'll put f(x) = sin(x)

Third part: graphs

Left: Absolute Value
Right: Linear

But in third part, it says “Name the parent function for each of the following graphs.” and shows two graphs.

Also, there are blanks labeled:

For first graph (left):
name: _______________
domain: _____________
range: ______________
asymptotes: _________
intercepts: _________
increasing/decreasing: _____
max/min: ____________
end behavior: ________
symmetry: ___________
transformations: _____

Similarly for second graph.

Oh! I missed that. In the image, after “Name the parent function...”, there are fields to fill for each graph.

So for each graph, we need to provide multiple properties.

Let’s do that.

First graph (left): V-shape, vertex at (0,0)

Parent function: Absolute Value → f(x) = |x|

Domain: all real numbers → (-∞, ∞)

Range: [0, ∞) (since absolute value is always ≥0)

Asymptotes: none (absolute value has no asymptotes)

Intercepts: x-intercept at (0,0), y-intercept at (0,0) → so (0,0)

Increasing/Decreasing: decreasing on (-∞, 0), increasing on (0, ∞)

Max/Min: minimum at (0,0), no maximum

End behavior: as x → ∞, f(x) → ∞; as x → -∞, f(x) → ∞

Symmetry: even function, symmetric about y-axis

Transformations: none (it’s the parent function)

Second graph (right): straight line through origin, slope 1 → f(x) = x

Parent function: Linear

Domain: (-∞, ∞)

Range: (-∞, ∞)

Asymptotes: none

Intercepts: (0,0)

Increasing/Decreasing: increasing everywhere (slope positive)

Max/Min: no max or min

End behavior: as x → ∞, f(x) → ∞; as x → -∞, f(x) → -∞

Symmetry: odd function, symmetric about origin

Transformations: none

Now, for the second part, “give an example...”, I need to write one equation.

I’ll use f(x) = sin(x)

But to match format, perhaps just write the expression.

Now, let’s organize final answers.

Since the user didn’t specify which parts to explain, and options are false, I’ll just provide the final answers as requested.

But according to rules, I must work through step by step first, which I did, and now provide Final Answer.

Final Answer should contain all required responses.

Structure:

For Part 1: list a to j

Part 2: one example

Part 3: for each graph, fill in the fields.

Since it’s text, I’ll write it clearly.

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Final Answer:

Part 1: Parent Function Names

a) Linear
b) Quadratic
c) Square Root
d) Absolute Value
e) Reciprocal
f) Cubic
g) Exponential
h) Logarithmic
i) Cube Root
j) Reciprocal Squared

Part 2: Example not covered
f(x) = sin(x) *(or any function not absolute value, cubic, logarithmic, or exponential)*

Part 3: Graph Analysis

*Left Graph (V-shaped):*
- Name: Absolute Value
- Domain: All real numbers or (-∞, ∞)
- Range: [0, ∞)
- Asymptotes: None
- Intercepts: (0, 0)
- Increasing/Decreasing: Decreasing on (-∞, 0); Increasing on (0, ∞)
- Max/Min: Minimum at (0, 0); No maximum
- End Behavior: As x → ∞, y → ∞; As x → -∞, y → ∞
- Symmetry: Even (symmetric about y-axis)
- Transformations: None (parent function)

*Right Graph (straight line through origin):*
- Name: Linear
- Domain: All real numbers or (-∞, ∞)
- Range: All real numbers or (-∞, ∞)
- Asymptotes: None
- Intercepts: (0, 0)
- Increasing/Decreasing: Increasing on (-∞, ∞)
- Max/Min: No maximum or minimum
- End Behavior: As x → ∞, y → ∞; As x → -∞, y → -∞
- Symmetry: Odd (symmetric about origin)
- Transformations: None (parent function)