Question

graph f(x) = -x² - 2x + 3 using graphing technology and use it to answer the questions.
2 matching 20 points
match the intervals where the function is:
increasing
decreasing
positive
negative
-3 < x < 1
x < -3 and x > 1
x > -1
x < -1

Explanation:

Step1: Find vertex of quadratic

The function is $f(x) = -x^2 - 2x + 3$, a downward-opening parabola. The x-coordinate of the vertex is $x = -\frac{b}{2a} = -\frac{-2}{2(-1)} = -1$.

Step2: Determine increasing/decreasing

For downward parabola, increasing left of vertex, decreasing right.

• Increasing: $x < -1$
• Decreasing: $x > -1$

Step3: Find x-intercepts

Set $-x^2 - 2x + 3 = 0$, factor to $-(x+3)(x-1)=0$. Intercepts at $x=-3, x=1$.

Step4: Determine positive/negative

Downward parabola is positive between intercepts, negative outside.

• Positive: $-3 < x < 1$
• Negative: $x < -3$ and $x > 1$

Answer:

• Increasing: $x < -1$
• Decreasing: $x > -1$
• Positive: $-3 < x < 1$
• Negative: $x < -3$ and $x > 1$