Sure, let's break down the quadratic function [tex]\( f(x) = -3x^2 + 6x + 5 \)[/tex] step-by-step to describe its graph accurately.
### 1. Opening Direction
First, we need to determine the direction in which the parabola opens. This is determined by the coefficient of the [tex]\( x^2 \)[/tex] term (also called the leading coefficient).
- The given quadratic function is [tex]\( f(x) = -3x^2 + 6x + 5 \)[/tex].
- The leading coefficient is -3.
If the leading coefficient is positive (greater than 0), the parabola opens upward. If the leading coefficient is negative (less than 0), the parabola opens downward.
Since -3 is less than 0, the parabola opens downward.
### 2. Stretching or Compressing
Next, we need to determine whether the parabola is stretched or compressed compared to the standard parabola [tex]\( f(x) = x^2 \)[/tex].
- We compare the absolute value of the leading coefficient to 1.
- The absolute value of -3 is [tex]\( | -3 | = 3 \)[/tex].
If the absolute value of the leading coefficient is greater than 1, the parabola is stretched (it becomes narrower). If the absolute value is less than 1, the parabola is compressed (it becomes wider). If the absolute value is exactly 1, the parabolic shape remains unchanged (neither stretched nor compressed).
Since 3 is greater than 1, the parabola is stretched.
### 3. Final Description
Combining these observations:
- The parabola opens downward.
- It is stretched by a factor of 3 (making it narrower than the standard parabola).
Therefore, the detailed graph description of the quadratic function [tex]\( f(x) = -3x^2 + 6x + 5 \)[/tex] is:
The parabola opens downward. It is stretched by a factor of 3.
