To sketch the graphs of the given functions, we will describe their characteristics and features step by step. 1. **Function: $ y = x $** - **Type:** Linear - **Graph:** A straight line passing through the origin (0,0) with a slope of 1. 2. **Function: $ y = x^2 $** - **Type:** Quadratic - **Graph:** A parabola that opens upwards with its vertex at the origin (0,0). The graph is symmetric about the y-axis. 3. **Function: $ y = x^3 $** - **Type:** Cubic - **Graph:** A curve that passes through the origin and has an inflection point at (0,0). It extends into the first quadrant for positive x and into the third quadrant for negative x. 4. **Function: $ y = |x| $** - **Type:** Absolute value function - **Graph:** A V-shaped graph that opens upwards with its vertex at the origin (0,0). It is symmetric about the y-axis. 5. **Function: $ y = \sqrt{x} $** - **Type:** Square root function - **Graph:** A curve that starts from the origin (0,0) and increases as x increases, existing only in the first quadrant. 6. **Function: $ y = \frac{1}{x} $** - **Type:** Rational (hyperbola) - **Graph:** Two branches: one in the first quadrant (for $ x 0 $) and one in the third quadrant (for $ x

Question

To sketch the graphs of the given functions, we will describe their characteristics and features step by step.

1. **Function: $ y = x $**
   - **Type:** Linear
   - **Graph:** A straight line passing through the origin (0,0) with a slope of 1.

2. **Function: $ y = x^2 $**
   - **Type:** Quadratic
   - **Graph:** A parabola that opens upwards with its vertex at the origin (0,0). The graph is symmetric about the y-axis.

3. **Function: $ y = x^3 $**
   - **Type:** Cubic
   - **Graph:** A curve that passes through the origin and has an inflection point at (0,0). It extends into the first quadrant for positive x and into the third quadrant for negative x.

4. **Function: $ y = |x| $**
   - **Type:** Absolute value function
   - **Graph:** A V-shaped graph that opens upwards with its vertex at the origin (0,0). It is symmetric about the y-axis.

5. **Function: $ y = \sqrt{x} $**
   - **Type:** Square root function
   - **Graph:** A curve that starts from the origin (0,0) and increases as x increases, existing only in the first quadrant.

6. **Function: $ y = \frac{1}{x} $**
   - **Type:** Rational (hyperbola)
   - **Graph:** Two branches: one in the first quadrant (for $ x > 0 $) and one in the third quadrant (for $ x < 0 $). There are asymptotes along the axes.

7. **Function: $ y = \frac{1}{x^2} $**
   - **Type:** Rational (hyperbola)
   - **Graph:** Curves that appear only in the first and second quadrants. There are vertical and horizontal asymptotes at the axes.

8. **Function: $ y = \sqrt{a^2 - x^2} $**
   - **Type:** Semi-circle
   - **Graph:** This represents the upper half of a circle with radius $ a $ centered at the origin. The graph exists where $ -a \leq x \leq a $.

### Graph Sketch Summary
- **Linear:** $ y = x $ is a diagonal line.
- **Quadratic:** $ y = x^2 $ is a U-shaped curve.
- **Cubic:** $ y = x^3 $ is an S-shaped curve.
- **Absolute Value:** $ y = |x| $ is a V-shaped graph.
- **Square Root:** $ y = \sqrt{x} $ is a curve starting from the origin.
- **Rational:** $ y = \frac{1}{x} $ and $ y = \frac{1}{x^2} $ are hyperbolas with asymptotes.
- **Semi-circle:** $ y = \sqrt{a^2 - x^2} $ is the upper half of a circle.

These descriptions should help you visualize the functions on a graph. For actual sketches, graphing software or graph paper can be used to represent these functions accurately.

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