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$Function worksheet.pdf - Math Models Name Worksheet 4.1 Relations ...$

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Show Answer Key & Explanations Step-by-step solution for: Function worksheet.pdf - Math Models Name Worksheet 4.1 Relations ...

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The image you provided is a worksheet titled "Parent Functions," which aims to help students understand and analyze the characteristics of different parent functions. The worksheet includes four types of parent functions: Linear, Absolute Value, Quadratic, and Cubic. Each function has its graph, equation, domain, range, intercepts, and other key characteristics listed.

Below, I will explain how to solve and fill in the missing information for each parent function:

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1. Linear Function

- Equation: $ y = x $
- Graph: A straight line passing through the origin (0, 0).
- Domain: All real numbers, $ (-\infty, \infty) $.
- Range: All real numbers, $ (-\infty, \infty) $.
- Intercepts:
- x-intercept: (0, 0)
- y-intercept: (0, 0)
- Characteristics:
- Slope: 1 (positive slope, increasing).
- No maximum or minimum value.
- Continuous and one-to-one function.

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2. Absolute Value Function

- Equation: $ y = |x| $
- Graph: V-shaped graph with its vertex at the origin (0, 0).
- Domain: All real numbers, $ (-\infty, \infty) $.
- Range: Non-negative real numbers, $ [0, \infty) $.
- Intercepts:
- x-intercept: (0, 0)
- y-intercept: (0, 0)
- Characteristics:
- Minimum value at $ (0, 0) $.
- Symmetric about the y-axis.
- Continuous but not one-to-one.

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3. Quadratic Function

- Equation: $ y = x^2 $
- Graph: Parabola opening upwards with its vertex at the origin (0, 0).
- Domain: All real numbers, $ (-\infty, \infty) $.
- Range: Non-negative real numbers, $ [0, \infty) $.
- Intercepts:
- x-intercept: (0, 0)
- y-intercept: (0, 0)
- Characteristics:
- Minimum value at $ (0, 0) $.
- Symmetric about the y-axis.
- Continuous but not one-to-one.

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4. Cubic Function

- Equation: $ y = x^3 $
- Graph: S-shaped curve passing through the origin (0, 0).
- Domain: All real numbers, $ (-\infty, \infty) $.
- Range: All real numbers, $ (-\infty, \infty) $.
- Intercepts:
- x-intercept: (0, 0)
- y-intercept: (0, 0)
- Characteristics:
- No maximum or minimum value.
- Odd function (symmetric about the origin).
- Continuous and one-to-one function.

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Summary of Key Characteristics

Here is a table summarizing the key characteristics for each function:

| Function | Equation | Domain | Range | x-intercept | y-intercept | Characteristics |
|-----------------|--------------|--------------------|--------------------|------------------|------------------|-------------------------------------------------------------------------------------|
| Linear | $ y = x $ | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ | (0, 0) | (0, 0) | Straight line, slope = 1, increasing, continuous, one-to-one. |
| Absolute Value | $ y = |x| $ | $ (-\infty, \infty) $ | $ [0, \infty) $ | (0, 0) | (0, 0) | V-shaped, minimum at (0, 0), symmetric about y-axis, continuous, not one-to-one. |
| Quadratic | $ y = x^2 $ | $ (-\infty, \infty) $ | $ [0, \infty) $ | (0, 0) | (0, 0) | Parabola opening up, minimum at (0, 0), symmetric about y-axis, continuous, not one-to-one. |
| Cubic | $ y = x^3 $ | $ (-\infty, \infty) $ | $ (-\infty, \infty) $ | (0, 0) | (0, 0) | S-shaped, odd function, symmetric about origin, continuous, one-to-one. |

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Final Answer

The completed worksheet should include the above details for each function. If there are specific sections in the worksheet that need filling in, use the information provided above to complete them accurately.

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Parent Tip: Review the logic above to help your child master the concept of math models worksheet.

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