Sure, let's analyze the quadratic function [tex]\( y = -x^2 + 4x + 3 \)[/tex] step-by-step.
### Step 1: Identify the coefficients.
The equation is given in the standard form [tex]\( y = ax^2 + bx + c \)[/tex].
Here:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = 3 \)[/tex]
### Step 2: Find the vertex of the parabola.
The vertex of a parabola described by [tex]\( y = ax^2 + bx + c \)[/tex] is given by the formula [tex]\( (h, k) \)[/tex], where:
- [tex]\( h = -\frac{b}{2a} \)[/tex]
- [tex]\( k = f(h) \)[/tex]
First, let's find [tex]\( h \)[/tex]:
[tex]\[ h = -\frac{b}{2a} = -\frac{4}{2 \cdot -1} = 2 \][/tex]
Next, substitute [tex]\( h \)[/tex] back into the function to find [tex]\( k \)[/tex]:
[tex]\[ k = -h^2 + 4h + 3 \][/tex]
[tex]\[ k = -(2)^2 + 4(2) + 3 \][/tex]
[tex]\[ k = -4 + 8 + 3 \][/tex]
[tex]\[ k = 7 \][/tex]
So, the vertex of the parabola is [tex]\( (2.0, 7.0) \)[/tex].
### Step 3: Find the roots of the quadratic equation.
The roots (or zeros) of the quadratic equation [tex]\( y = -x^2 + 4x + 3 \)[/tex] can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = 4^2 - 4(-1)(3) \][/tex]
[tex]\[ \text{Discriminant} = 16 + 12 \][/tex]
[tex]\[ \text{Discriminant} = 28 \][/tex]
Since the discriminant is positive, we have two real roots.
Now, apply the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{28}}{2(-1)} \][/tex]
[tex]\[ x = \frac{-4 \pm \sqrt{28}}{-2} \][/tex]
[tex]\[ x = \frac{-4 \pm 2\sqrt{7}}{-2} \][/tex]
This simplifies to:
[tex]\[ x = \frac{4 \mp 2\sqrt{7}}{2} \][/tex]
[tex]\[ x = 2 \mp \sqrt{7} \][/tex]
So, the roots are:
[tex]\[ x_1 = 2 - \sqrt{7} \approx -0.6457513110645907 \][/tex]
[tex]\[ x_2 = 2 + \sqrt{7} \approx 4.645751311064591 \][/tex]
### Summary:
- The vertex of the parabola is at [tex]\( (2.0, 7.0) \)[/tex].
- The roots of the quadratic equation are approximately [tex]\( x_1 \approx -0.6457513110645907 \)[/tex] and [tex]\( x_2 \approx 4.645751311064591 \)[/tex].
