Comparison of linear, quadratic, and exponential functions with graphs and characteristics.

Let's solve this step-by-step by filling in the table with the domain, range, and other characteristics for each type of function: Linear, Quadratic, and Exponential, based on their graphs and parent functions.

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🔹 1. LINEAR FUNCTION

- Parent Function: $ f(x) = x $

#### ✔ Domain:
All real numbers — you can plug any real number into $ x $.
> Domain: $ (-\infty, \infty) $

#### ✔ Range:
The output $ y $ can also be any real number since the line goes infinitely in both directions.
> Range: $ (-\infty, \infty) $

#### ✔ Other Characteristics:
- The graph is a straight line passing through the origin (0, 0).
- It has a constant rate of change (slope = 1).
- It is increasing everywhere.
- No maximum or minimum value.
- Symmetric about the line $ y = x $ (but not symmetric in the traditional sense like parabolas).

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🔹 2. QUADRATIC FUNCTION

- Parent Function: $ f(x) = x^2 $

#### ✔ Domain:
Any real number can be squared.
> Domain: $ (-\infty, \infty) $

#### ✔ Range:
Since squaring any real number gives a non-negative result, the smallest value is 0 at $ x = 0 $. So $ y \geq 0 $.
> Range: $ [0, \infty) $

#### ✔ Other Characteristics:
- The graph is a parabola opening upward.
- Vertex at $ (0, 0) $ — this is the minimum point.
- Symmetric about the y-axis (axis of symmetry: $ x = 0 $).
- Decreases on $ (-\infty, 0) $, increases on $ (0, \infty) $.
- Continuous and smooth.

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🔹 3. EXPONENTIAL FUNCTION

- Parent Function: $ f(x) = 2^x $

#### ✔ Domain:
You can raise 2 to any real power (positive, negative, zero), so all real numbers are allowed.
> Domain: $ (-\infty, \infty) $

#### ✔ Range:
$ 2^x > 0 $ for all $ x $, but it never reaches 0. As $ x \to -\infty $, $ 2^x \to 0 $, but never touches it.
> Range: $ (0, \infty) $

#### ✔ Other Characteristics:
- The graph is always increasing (since base $ 2 > 1 $).
- Has a horizontal asymptote at $ y = 0 $ (the x-axis).
- Passes through $ (0, 1) $ because $ 2^0 = 1 $.
- Growth becomes faster as $ x $ increases (exponential growth).
- Not symmetric.
- Continuous and smooth.

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✔ Final Filled Table:

| | LINEAR | QUADRATIC | EXPONENTIAL |
|------------------|--------------------------|----------------------------|------------------------------|
| GRAPH | Straight line through origin | U-shaped parabola | Curve rising rapidly to right, approaching x-axis on left |
| PARENT FUNCTION | $ f(x) = x $ | $ f(x) = x^2 $ | $ f(x) = 2^x $ |
| DOMAIN | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ |
| RANGE | $ (-\infty, \infty) $ | $ [0, \infty) $ | $ (0, \infty) $ |
| OTHER CHARACTERISTICS | - Straight line<br>- Constant slope<br>- Increases everywhere<br>- No max/min | - Parabola opens up<br>- Vertex at (0,0)<br>- Minimum at vertex<br>- Symmetric about y-axis | - Always increasing<br>- Horizontal asymptote: $ y = 0 $<br>- Passes through (0,1)<br>- Exponential growth |

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📌 Summary of Key Differences & Similarities

| Feature | Linear | Quadratic | Exponential |
|----------------------|---------------------------|-----------------------------|-----------------------------|
| Shape | Straight line | Parabola | Curved (grows fast) |
| Rate of Change | Constant | Changes (increases/decreases) | Increases over time |
| Asymptotes | None | None | Yes (horizontal: $ y=0 $) |
| Symmetry | None (unless reflected) | Yes (about y-axis) | No |
| Max/Min | None | Minimum at vertex | No max/min (always increasing) |

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✔ This comparison helps students understand how different types of functions behave and how their algebraic forms relate to their graphs.

Let me know if you'd like this formatted as a printable worksheet!