https://file.aiquickdraw.com/mathbot/file/1725851019457y5u13kj7.jpg

To find the functions represented by the graph, we need to identify the two curves. The graph depicts two functions: a quadratic function \( f(x) \) and a linear function. Let's analyze the graph step-by-step.

### Step 1: Identify the Linear Function

The magenta line is a straight line, so it can be represented by the equation:
\[ y = mx + b \]

From the graph, we can identify two points that the line passes through:
1. \( (0, 0.5) \)
2. \( (3, 2.5) \)

Using these points, we can find the slope \( m \):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2.5 - 0.5}{3 - 0} = \frac{2}{3} \]

Now we use the slope-intercept form to find \( b \). Using the point \( (0, 0.5) \):
\[ 0.5 = \frac{2}{3}(0) + b \implies b = 0.5 \]

Thus, the equation of the line is:
\[ y = \frac{2}{3}x + 0.5 \]

### Step 2: Identify the Quadratic Function

The blue curve appears to be a quadratic function of the form:
\[ y = ax^2 + bx + c \]

From the graph, we can identify a few points to help us find the coefficients \( a \), \( b \), and \( c \):
1. \( (0, 2) \)
2. \( (1, 1.5) \)
3. \( (3, 3) \)

We now set up a system of equations using these points:

1. At \( (0, 2) \):
\[ 2 = a(0)^2 + b(0) + c \implies c = 2 \]

2. At \( (1, 1.5) \):
\[ 1.5 = a(1)^2 + b(1) + 2 \implies 1.5 = a + b + 2 \implies a + b = -0.5 \]

3. At \( (3, 3) \):
\[ 3 = a(3)^2 + b(3) + 2 \implies 3 = 9a + 3b + 2 \implies 9a + 3b = 1 \]

We now have the following system of linear equations:
1. \( c = 2 \)
2. \( a + b = -0.5 \)
3. \( 9a + 3b = 1 \)

We can solve this system to find \( a \) and \( b \).

From \( a + b = -0.5 \):
\[ b = -0.5 - a \]

Substitute \( b \) into \( 9a + 3b = 1 \):
\[ 9a + 3(-0.5 - a) = 1 \]
\[ 9a - 1.5 - 3a = 1 \]
\[ 6a - 1.5 = 1 \]
\[ 6a = 2.5 \]
\[ a = \frac{2.5}{6} = \frac{5}{12} \]

Now substitute \( a \) back into \( b = -0.5 - a \):
\[ b = -0.5 - \frac{5}{12} \]
\[ b = -\frac{6}{12} - \frac{5}{12} \]
\[ b = -\frac{11}{12} \]

So the quadratic function is:
\[ y = \frac{5}{12}x^2 - \frac{11}{12}x + 2 \]

### Summary

- Linear function: \( y = \frac{2}{3}x + 0.5 \)
- Quadratic function: \( y = \frac{5}{12}x^2 - \frac{11}{12}x + 2 \)

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would you find the intersection points of these two functions algebraically?
2. What are the vertex and axis of symmetry for the quadratic function?
3. How would you verify these functions by plotting them on a graph?
4. What is the significance of the slope and y-intercept in the context of the linear function?
5. How does changing the coefficients \( a \), \( b \), and \( c \) affect the shape of the quadratic function?

**Tip:** When working with systems of equations, always check your solutions by substituting them back into the original equations to ensure they satisfy all conditions.

Can you find the function from this graph?

Discover how to find the linear and quadratic functions represented by the graph. Using key points and formulas, learn how to calculate the equations for each curve, including the slope, y-intercept, and solving systems of equations. This problem is ideal for students in Grades 9-12.

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