Certainly! Let's identify the properties of the quadratic equation given by [tex]\( y = -3x^2 + 6x + 17 \)[/tex].
### Step 1: Identify the coefficients
The general form of a quadratic equation is [tex]\( y = ax^2 + bx + c \)[/tex].
In this equation:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = 17 \)[/tex]
### Step 2: Calculate the vertex
The vertex [tex]\((h, k)\)[/tex] of a quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formulas:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
[tex]\[
k = a \cdot h^2 + b \cdot h + c
\][/tex]
For our given equation:
[tex]\[
h = -\frac{6}{2 \cdot (-3)} = -\frac{6}{-6} = 1
\][/tex]
Next, we find [tex]\( k \)[/tex]:
[tex]\[
k = -3 \cdot (1)^2 + 6 \cdot 1 + 17 = -3 \cdot 1 + 6 \cdot 1 + 17 = -3 + 6 + 17 = 20
\][/tex]
So, the vertex is [tex]\((1.0, 20.0)\)[/tex].
### Step 3: Axis of symmetry
The axis of symmetry for a quadratic equation is given by the line [tex]\( x = h \)[/tex].
For our equation:
[tex]\[
x = 1
\][/tex]
### Step 4: Determine the direction of opening
The direction in which the parabola opens depends on the coefficient [tex]\( a \)[/tex]:
- If [tex]\( a \)[/tex] is positive, the parabola opens upwards.
- If [tex]\( a \)[/tex] is negative, the parabola opens downwards.
For our equation, [tex]\( a = -3 \)[/tex], which is negative. Thus, the parabola opens downwards.
### Summary
- Vertex (h, k): [tex]\((1.0, 20.0)\)[/tex]
- Axis of symmetry: [tex]\( x = 1 \)[/tex]
- Direction of opening: Downwards
By carefully following these steps, we have identified the key properties of the quadratic equation [tex]\( y = -3x^2 + 6x + 17 \)[/tex].
