https://file.aiquickdraw.com/mathbot/file/17266871267968lhxx7z8.jpg

The image you provided shows four different equations, each representing different types of functions:

**a. \( y = x^2 \):**
- This is a quadratic function. It represents a parabola that opens upwards, with the vertex at the origin (0,0).

**b. \( y = x + 3 \):**
- This is a linear function with a slope of 1 and a y-intercept of 3. The graph is a straight line that crosses the y-axis at \( y = 3 \).

**c. \( y = 2^x \):**
- This is an exponential function. It grows exponentially as \( x \) increases, with a base of 2. The graph passes through the point (0,1) and increases rapidly as \( x \) becomes larger.

**d. \( f(x) = x(x + 2) \):**
- This is another quadratic function, but in expanded form it is \( f(x) = x^2 + 2x \). It also represents a parabola, and its vertex and roots can be determined by solving the equation.

Would you like more detailed explanations of any specific graph or function? Or perhaps you're interested in some visualizations of these functions?

Here are five related questions to expand your understanding:
1. What is the vertex of the quadratic function in part **d**?
2. How do you find the x-intercept of the linear function in part **b**?
3. What are the properties of exponential growth, as seen in part **c**?
4. How does the shape of the parabola in part **a** change if the equation is altered to \( y = -x^2 \)?
5. What are the solutions to the equation \( f(x) = 0 \) in part **d**?

**Tip:** For quadratic functions, the vertex can be found using the formula \( x = -\frac{b}{2a} \) when written in standard form \( ax^2 + bx + c \).

Identify the types of functions represented by the following equations: y = x^2, y = x + 3, y = 2^x, f(x) = x(x + 2).

Learn about quadratic, linear, and exponential functions through equations such as y = x^2, y = x + 3, y = 2^x, and f(x) = x(x + 2). Explore key algebraic concepts with step-by-step explanations for students in Grades 8-10.

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