Matching graphs to equations worksheet for algebra.

To solve the problem of matching the graphs with their corresponding equations, we need to analyze the shape and behavior of each graph and compare it with the characteristics of the given equations. Let's go through each graph and equation step by step.

Graph Analysis and Matching

#### Row 1:
1. First Graph (Top Left):
- Shape: A U-shaped curve that opens upwards.
- Behavior: Symmetric around the y-axis.
- Matches: This is a quadratic function that opens upwards.
- Equation: \( y = x^2 + x \) (E)

2. Second Graph (Top Middle):
- Shape: A U-shaped curve that opens upwards.
- Behavior: Symmetric around the y-axis.
- Matches: This is a quadratic function that opens upwards.
- Equation: \( y = x^2 - x \) (B)

3. Third Graph (Top Right):
- Shape: An S-shaped curve (cubic).
- Behavior: Changes direction twice.
- Matches: This is a cubic function with a positive leading coefficient.
- Equation: \( y = x^3 - x^2 + 1 \) (C)

#### Row 2:
4. Fourth Graph (Middle Left):
- Shape: A hyperbola in the second and fourth quadrants.
- Behavior: As \( x \to 0 \), \( y \to \infty \); as \( x \to \infty \), \( y \to 0 \).
- Matches: This is a reciprocal function with a negative sign.
- Equation: \( y = -\frac{3}{x} \) (D)

5. Fifth Graph (Middle Middle):
- Shape: An exponential curve that increases rapidly.
- Behavior: As \( x \to \infty \), \( y \to \infty \); as \( x \to -\infty \), \( y \to 0 \).
- Matches: This is an exponential function with base greater than 1.
- Equation: \( y = 2^x \) (G)

6. Sixth Graph (Middle Right):
- Shape: A U-shaped curve that opens upwards.
- Behavior: Symmetric around the y-axis.
- Matches: This is a quadratic function that opens upwards.
- Equation: \( y = 2x^2 - x^3 \) (H)

#### Row 3:
7. Seventh Graph (Bottom Left):
- Shape: An S-shaped curve (cubic).
- Behavior: Changes direction twice.
- Matches: This is a cubic function with a positive leading coefficient.
- Equation: \( y = x^3 - x^2 \) (I)

8. Eighth Graph (Bottom Middle):
- Shape: A parabola that opens downwards.
- Behavior: Symmetric around the y-axis.
- Matches: This is a quadratic function that opens downwards.
- Equation: \( y = x - x^2 \) (J)

9. Ninth Graph (Bottom Right):
- Shape: A hyperbola in the first and third quadrants.
- Behavior: As \( x \to 0 \), \( y \to \infty \); as \( x \to \infty \), \( y \to 0 \).
- Matches: This is a reciprocal function with a positive sign.
- Equation: \( y = \frac{3}{x} \) (F)

#### Row 4:
10. Tenth Graph (Bottom Left):
- Shape: An exponential curve that decreases rapidly.
- Behavior: As \( x \to \infty \), \( y \to 0 \); as \( x \to -\infty \), \( y \to \infty \).
- Matches: This is an exponential function with base less than 1.
- Equation: \( y = 2^{-x} \) (A)

11. Eleventh Graph (Bottom Middle):
- Shape: An exponential curve that increases rapidly.
- Behavior: As \( x \to \infty \), \( y \to \infty \); as \( x \to -\infty \), \( y \to 0 \).
- Matches: This is an exponential function with base greater than 1.
- Equation: \( y = -2^x \) (L)

12. Twelfth Graph (Bottom Right):
- Shape: An S-shaped curve (cubic).
- Behavior: Changes direction twice.
- Matches: This is a cubic function with a negative leading coefficient.
- Equation: \( y = -x^3 - x^2 + 1 \) (K)

Final Matching:

- Top Left: \( y = x^2 + x \) (E)
- Top Middle: \( y = x^2 - x \) (B)
- Top Right: \( y = x^3 - x^2 + 1 \) (C)
- Middle Left: \( y = -\frac{3}{x} \) (D)
- Middle Middle: \( y = 2^x \) (G)
- Middle Right: \( y = 2x^2 - x^3 \) (H)
- Bottom Left: \( y = x^3 - x^2 \) (I)
- Bottom Middle: \( y = x - x^2 \) (J)
- Bottom Right: \( y = \frac{3}{x} \) (F)
- Bottom Left: \( y = 2^{-x} \) (A)
- Bottom Middle: \( y = -2^x \) (L)
- Bottom Right: \( y = -x^3 - x^2 + 1 \) (K)

Answer:

\[
\boxed{
\begin{array}{ccc}
\text{Top Left} & \text{Top Middle} & \text{Top Right} \\
E & B & C \\
\text{Middle Left} & \text{Middle Middle} & \text{Middle Right} \\
D & G & H \\
\text{Bottom Left} & \text{Bottom Middle} & \text{Bottom Right} \\
I & J & F \\
\text{Bottom Left} & \text{Bottom Middle} & \text{Bottom Right} \\
A & L & K \\
\end{array}
}
\]