Question

Describe the intervals of increasing/decreasing of the following quadratic function graphed.
(2,0) (3,1) (1,0)


(4 points)
The quadratic function is increasing from
to
and decreasing from
to
.

Answer 1

To analyze the intervals of increasing and decreasing for the given points, we first need to determine the quadratic function that fits the points (2,0), (3,1), and (1,0).

The general form of a quadratic function is:
\[ f(x) = ax^2 + bx + c \]

Given the points:
1. (1, 0): This gives us the equation \( a(1)^2 + b(1) + c = 0 \).
2. (2, 0): This gives us the equation \( a(2)^2 + b(2) + c = 0 \).
3. (3, 1): This gives us the equation \( a(3)^2 + b(3) + c = 1 \).

From these points, we can determine that the function has roots at \( x = 1 \) and \( x = 2 \), which means \( f(x) = a(x-1)(x-2) \).

To find the value of \( a \), we can substitute the third point (3,1) into our quadratic function:
\[ f(3) = a(3-1)(3-2) = 1 \]
\[ 2a = 1 \]
\[ a = \frac{1}{2} \]

So the quadratic function is:
\[ f(x) = \frac{1}{2}(x-1)(x-2) \]

Expanding this:
\[ f(x) = \frac{1}{2}(x^2 - 3x + 2) = \frac{1}{2}x^2 - \frac{3}{2}x + 1 \]

To find the intervals of increasing and decreasing, we find the vertex of the parabola using the formula:
\[ x = -\frac{b}{2a} \]
Here, \( a = \frac{1}{2} \) and \( b = -\frac{3}{2} \):
\[ x = -\frac{-\frac{3}{2}}{2 \cdot \frac{1}{2}} = \frac{3}{2} \]

The vertex \( x = \frac{3}{2} \) is the point where the function changes from decreasing to increasing.

Now, we determine the intervals:
- The function is **decreasing** on the interval \( (-\infty, \frac{3}{2}) \).
- The function is **increasing** on the interval \( (\frac{3}{2}, \infty) \).

### Summary:
- The quadratic function is increasing from **\( \frac{3}{2} \) to \( \infty \)** and decreasing from **\( -\infty \) to \( \frac{3}{2} \)**.