Certainly! Let's analyze the quadratic function [tex]\( y = x^2 - 8x + 15 \)[/tex] in detail.
### Step 1: Identify the vertex
The quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex] has its vertex at [tex]\( x = -\frac{b}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 15 \)[/tex].
[tex]\[ x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \][/tex]
So, the x-coordinate of the vertex is [tex]\( x = 4 \)[/tex].
To find the y-coordinate of the vertex, we substitute [tex]\( x = 4 \)[/tex] back into the equation:
[tex]\[ y = (4)^2 - 8(4) + 15 \][/tex]
[tex]\[ y = 16 - 32 + 15 \][/tex]
[tex]\[ y = -1 \][/tex]
Thus, the vertex of the function is [tex]\( (4, -1) \)[/tex].
### Step 2: Find the roots of the equation
To find the roots of the quadratic function, we need to solve the equation [tex]\( x^2 - 8x + 15 = 0 \)[/tex].
### Step 3: Factorize the quadratic expression
We can factorize [tex]\( x^2 - 8x + 15 \)[/tex]:
[tex]\[ x^2 - 8x + 15 = (x - 3)(x - 5) \][/tex]
### Step 4: Solve for the roots
Setting each factor equal to zero gives us the roots of the quadratic equation:
[tex]\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \][/tex]
[tex]\[ x - 5 = 0 \quad \Rightarrow \quad x = 5 \][/tex]
So, the roots of the equation are [tex]\( x = 3 \)[/tex] and [tex]\( x = 5 \)[/tex].
### Summary
- Vertex: The vertex of the quadratic function [tex]\( y = x^2 - 8x + 15 \)[/tex] is at [tex]\( (4, -1) \)[/tex].
- Roots: The roots of the equation [tex]\( x^2 - 8x + 15 = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = 5 \)[/tex].
So, putting it all together:
- The vertex of the function is: [tex]\( (4, -1) \)[/tex]
- The roots of the equation are: [tex]\( x = 3 \)[/tex] and [tex]\( x = 5 \)[/tex]
