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To answer the question regarding the graph of the function [tex]\( f(x) = x^2 - 6x + 5 \)[/tex], we need to analyze its key features and shape. Here's a step-by-step detailed solution:
### Step 1: Identify the Type of Function
The given function [tex]\( f(x) = x^2 - 6x + 5 \)[/tex] is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex]. Quadratic functions graph as parabolas.
### Step 2: Determine the Direction of the Parabola
Since the coefficient of [tex]\( x^2 \)[/tex] (which is 1) is positive, the parabola opens upwards.
### Step 3: Find the Vertex
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex]:
[tex]\[ x = -\frac{-6}{2 \cdot 1} = \frac{6}{2} = 3 \][/tex]
To find the [tex]\( y \)[/tex]-coordinate of the vertex, substitute [tex]\( x = 3 \)[/tex] back into the function:
[tex]\[ f(3) = 3^2 - 6 \cdot 3 + 5 \][/tex]
[tex]\[ f(3) = 9 - 18 + 5 \][/tex]
[tex]\[ f(3) = -4 \][/tex]
Thus, the vertex is at [tex]\( (3, -4) \)[/tex].
### Step 4: Find the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^2 - 6 \cdot 0 + 5 \][/tex]
[tex]\[ f(0) = 5 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 5) \)[/tex].
### Step 5: Find the Roots (or [tex]\( x \)[/tex]-intercepts)
The roots of the quadratic function are found by solving [tex]\( x^2 - 6x + 5 = 0 \)[/tex]. We can use factoring for this:
[tex]\[ x^2 - 6x + 5 = (x-1)(x-5) = 0 \][/tex]
So the roots are:
[tex]\[ x = 1 \][/tex]
[tex]\[ x = 5 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts are at [tex]\( (1, 0) \)[/tex] and [tex]\( (5, 0) \)[/tex].
### Step 6: Sketch the Graph
Given all the key points and the direction of the parabola:
- The vertex is at [tex]\( (3, -4) \)[/tex].
- The [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 5) \)[/tex].
- The roots are [tex]\( (1, 0) \)[/tex] and [tex]\( (5, 0) \)[/tex].
- The parabola opens upwards.
Now, using these points and features, we can sketch the graph:
1. Start at the vertex [tex]\( (3, -4) \)[/tex], the lowest point on the graph.
2. The curve goes through [tex]\( (1, 0) \)[/tex] and [tex]\( (5, 0) \)[/tex].
3. It crosses the [tex]\( y \)[/tex]-axis at [tex]\( (0, 5) \)[/tex].
4. Since the parabola opens upward, it curves upwards from the vertex moving through these intercepts.
The accurate graph should resemble a 'U'-shaped curve (a parabola) that indicates the function's specific key features and direction.
This detailed consideration provides a clear understanding of how to draw the graph of [tex]\( f(x) = x^2 - 6x + 5 \)[/tex].
### Step 1: Identify the Type of Function
The given function [tex]\( f(x) = x^2 - 6x + 5 \)[/tex] is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex]. Quadratic functions graph as parabolas.
### Step 2: Determine the Direction of the Parabola
Since the coefficient of [tex]\( x^2 \)[/tex] (which is 1) is positive, the parabola opens upwards.
### Step 3: Find the Vertex
The vertex of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex]:
[tex]\[ x = -\frac{-6}{2 \cdot 1} = \frac{6}{2} = 3 \][/tex]
To find the [tex]\( y \)[/tex]-coordinate of the vertex, substitute [tex]\( x = 3 \)[/tex] back into the function:
[tex]\[ f(3) = 3^2 - 6 \cdot 3 + 5 \][/tex]
[tex]\[ f(3) = 9 - 18 + 5 \][/tex]
[tex]\[ f(3) = -4 \][/tex]
Thus, the vertex is at [tex]\( (3, -4) \)[/tex].
### Step 4: Find the [tex]\( y \)[/tex]-intercept
The [tex]\( y \)[/tex]-intercept occurs when [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0^2 - 6 \cdot 0 + 5 \][/tex]
[tex]\[ f(0) = 5 \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 5) \)[/tex].
### Step 5: Find the Roots (or [tex]\( x \)[/tex]-intercepts)
The roots of the quadratic function are found by solving [tex]\( x^2 - 6x + 5 = 0 \)[/tex]. We can use factoring for this:
[tex]\[ x^2 - 6x + 5 = (x-1)(x-5) = 0 \][/tex]
So the roots are:
[tex]\[ x = 1 \][/tex]
[tex]\[ x = 5 \][/tex]
Thus, the [tex]\( x \)[/tex]-intercepts are at [tex]\( (1, 0) \)[/tex] and [tex]\( (5, 0) \)[/tex].
### Step 6: Sketch the Graph
Given all the key points and the direction of the parabola:
- The vertex is at [tex]\( (3, -4) \)[/tex].
- The [tex]\( y \)[/tex]-intercept is at [tex]\( (0, 5) \)[/tex].
- The roots are [tex]\( (1, 0) \)[/tex] and [tex]\( (5, 0) \)[/tex].
- The parabola opens upwards.
Now, using these points and features, we can sketch the graph:
1. Start at the vertex [tex]\( (3, -4) \)[/tex], the lowest point on the graph.
2. The curve goes through [tex]\( (1, 0) \)[/tex] and [tex]\( (5, 0) \)[/tex].
3. It crosses the [tex]\( y \)[/tex]-axis at [tex]\( (0, 5) \)[/tex].
4. Since the parabola opens upward, it curves upwards from the vertex moving through these intercepts.
The accurate graph should resemble a 'U'-shaped curve (a parabola) that indicates the function's specific key features and direction.
This detailed consideration provides a clear understanding of how to draw the graph of [tex]\( f(x) = x^2 - 6x + 5 \)[/tex].
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