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To solve the function [tex]\( f(x) = (x-1)(x-7) \)[/tex], we will begin by understanding and simplifying it step by step. Here’s a detailed breakdown:
### Step 1: Understand the Function
The given function [tex]\( f(x) = (x-1)(x-7) \)[/tex] is a product of two linear terms, [tex]\( (x-1) \)[/tex] and [tex]\( (x-7) \)[/tex].
### Step 2: Expand the Expression
To better understand the behavior of the function, we can expand this product. Here's how to expand it:
[tex]\[ f(x) = (x-1)(x-7) \][/tex]
We use the distributive property (also known as FOIL in some contexts for binomials):
[tex]\[ f(x) = x \cdot x - x \cdot 7 - 1 \cdot x + 1 \cdot 7 \][/tex]
Simplify each term:
[tex]\[ f(x) = x^2 - 7x - x + 7 \][/tex]
Combine like terms:
[tex]\[ f(x) = x^2 - 8x + 7 \][/tex]
### Step 3: Identify Key Features of the Function
#### The Roots:
To find the roots (or zeros) of the function, we set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ (x-1)(x-7) = 0 \][/tex]
Setting each factor to zero gives us the roots:
[tex]\[ x - 1 = 0 \implies x = 1 \][/tex]
[tex]\[ x - 7 = 0 \implies x = 7 \][/tex]
So, the function [tex]\( f(x) \)[/tex] has roots at [tex]\( x = 1 \)[/tex] and [tex]\( x = 7 \)[/tex].
#### Vertex:
Since the function is a quadratic in standard form [tex]\( ax^2 + bx + c \)[/tex], we can find the vertex. The [tex]\( x \)[/tex]-coordinate of the vertex for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -8 \)[/tex], so:
[tex]\[ x = -\frac{-8}{2 \times 1} = \frac{8}{2} = 4 \][/tex]
Substitute [tex]\( x = 4 \)[/tex] back into the function to find the [tex]\( y \)[/tex]-coordinate of the vertex:
[tex]\[ f(4) = (4-1)(4-7) = 3 \cdot -3 = -9 \][/tex]
So, the vertex of the parabola is at [tex]\( (4, -9) \)[/tex].
#### Direction and Width of the Parabola:
The coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 1 \)[/tex]) is positive, so the parabola opens upwards. The standard quadratic form also indicates the parabola has a typical 'standard' width.
### Step 4: Sketch the Function
When sketching [tex]\( f(x) = (x-1)(x-7) \)[/tex] or [tex]\( f(x) = x^2 - 8x + 7 \)[/tex]:
- Plot the roots at [tex]\( x = 1 \)[/tex] and [tex]\( x = 7 \)[/tex].
- Plot the vertex at [tex]\( (4, -9) \)[/tex].
- Because the parabola opens upwards, draw a smooth curve through these points with the vertex as the lowest point.
By examining the form of the function and following these steps systematically, we have a clear understanding of [tex]\( f(x) \)[/tex] and can sketch or describe its behavior.
### Step 1: Understand the Function
The given function [tex]\( f(x) = (x-1)(x-7) \)[/tex] is a product of two linear terms, [tex]\( (x-1) \)[/tex] and [tex]\( (x-7) \)[/tex].
### Step 2: Expand the Expression
To better understand the behavior of the function, we can expand this product. Here's how to expand it:
[tex]\[ f(x) = (x-1)(x-7) \][/tex]
We use the distributive property (also known as FOIL in some contexts for binomials):
[tex]\[ f(x) = x \cdot x - x \cdot 7 - 1 \cdot x + 1 \cdot 7 \][/tex]
Simplify each term:
[tex]\[ f(x) = x^2 - 7x - x + 7 \][/tex]
Combine like terms:
[tex]\[ f(x) = x^2 - 8x + 7 \][/tex]
### Step 3: Identify Key Features of the Function
#### The Roots:
To find the roots (or zeros) of the function, we set [tex]\( f(x) = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[ (x-1)(x-7) = 0 \][/tex]
Setting each factor to zero gives us the roots:
[tex]\[ x - 1 = 0 \implies x = 1 \][/tex]
[tex]\[ x - 7 = 0 \implies x = 7 \][/tex]
So, the function [tex]\( f(x) \)[/tex] has roots at [tex]\( x = 1 \)[/tex] and [tex]\( x = 7 \)[/tex].
#### Vertex:
Since the function is a quadratic in standard form [tex]\( ax^2 + bx + c \)[/tex], we can find the vertex. The [tex]\( x \)[/tex]-coordinate of the vertex for a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -8 \)[/tex], so:
[tex]\[ x = -\frac{-8}{2 \times 1} = \frac{8}{2} = 4 \][/tex]
Substitute [tex]\( x = 4 \)[/tex] back into the function to find the [tex]\( y \)[/tex]-coordinate of the vertex:
[tex]\[ f(4) = (4-1)(4-7) = 3 \cdot -3 = -9 \][/tex]
So, the vertex of the parabola is at [tex]\( (4, -9) \)[/tex].
#### Direction and Width of the Parabola:
The coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( a = 1 \)[/tex]) is positive, so the parabola opens upwards. The standard quadratic form also indicates the parabola has a typical 'standard' width.
### Step 4: Sketch the Function
When sketching [tex]\( f(x) = (x-1)(x-7) \)[/tex] or [tex]\( f(x) = x^2 - 8x + 7 \)[/tex]:
- Plot the roots at [tex]\( x = 1 \)[/tex] and [tex]\( x = 7 \)[/tex].
- Plot the vertex at [tex]\( (4, -9) \)[/tex].
- Because the parabola opens upwards, draw a smooth curve through these points with the vertex as the lowest point.
By examining the form of the function and following these steps systematically, we have a clear understanding of [tex]\( f(x) \)[/tex] and can sketch or describe its behavior.
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