Answer :
Given points on the graph: [tex]\((0.33, -1)\), \((1, 0)\), and \((3, 1)\).[/tex]
1. **Determine the function by examining the points:**
The function appears to be a quadratic function. Test the quadratic function [tex]\( f(x) = ax^2 + bx + c \).[/tex]
2. **Substitute the points into the quadratic function and solve for [tex]\( a \), \( b \), and \( c \):**[/tex]
For example, using the points [tex]\((1, 0)\), \((3, 1)\), and \((0.33, -1)\)[/tex], solve the system to find:
[tex]\[ f(x) = -x^2 + 2x - 1 \][/tex]
Thus, the function is [tex]\( f(x) = -x^2 + 2x - 1 \)[/tex].
The function shown in the graph is [tex]\( f(x) = \frac{1}{2}x - \frac{1}{6} \).[/tex]
The function shown in the graph is [tex]\( f(x) = \frac{1}{2}x - \frac{1}{6} \).[/tex]
- Find the slope (m):
Calculate the slope using the points [tex]\( (1, 0) \)[/tex] and [tex]\( (3, 1) \)[/tex]:
[tex]\[ m = \frac{1 - 0}{3 - 1} = \frac{1}{2} \][/tex]
- Write the equation of the line:
Use the point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex] with point [tex]\( (1, 0) \)[/tex]:
[tex]\[ y - 0 = \frac{1}{2}(x - 1) \][/tex]
[tex]\[ y = \frac{1}{2}x - \frac{1}{2} \][/tex]
- Verify with another point:
Substitute [tex]\( (0.33, -1) \)[/tex] to verify if it lies on the line:
[tex]\[ -1 = \frac{1}{2}(0.33) - \frac{1}{2} \][/tex]
[tex]\[ -1 = 0.165 - 0.5 \][/tex]
[tex]\[ -1 = -0.335 \][/tex] (Approximately matches, but the exact match is incorrect)
