Let's analyze and sketch the function [tex]\( f(x) = x - 5 \)[/tex] step-by-step.
1. Points of the Function:
To graph [tex]\( f(x) = x - 5 \)[/tex], we'll start by calculating several points that lie on the line. These points are:
- For [tex]\( x = -5 \)[/tex]: [tex]\( f(-5) = -5 - 5 = -10 \)[/tex]
- For [tex]\( x = -4 \)[/tex]: [tex]\( f(-4) = -4 - 5 = -9 \)[/tex]
- For [tex]\( x = -3 \)[/tex]: [tex]\( f(-3) = -3 - 5 = -8 \)[/tex]
- For [tex]\( x = -2 \)[/tex]: [tex]\( f(-2) = -2 - 5 = -7 \)[/tex]
- For [tex]\( x = -1 \)[/tex]: [tex]\( f(-1) = -1 - 5 = -6 \)[/tex]
- For [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = 0 - 5 = -5 \)[/tex]
- For [tex]\( x = 1 \)[/tex]: [tex]\( f(1) = 1 - 5 = -4 \)[/tex]
- For [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = 2 - 5 = -3 \)[/tex]
- For [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = 3 - 5 = -2 \)[/tex]
- For [tex]\( x = 4 \)[/tex]: [tex]\( f(4) = 4 - 5 = -1 \)[/tex]
- For [tex]\( x = 5 \)[/tex]: [tex]\( f(5) = 5 - 5 = 0 \)[/tex]
Therefore, the points on the graph are:
[tex]\[ (-5,-10), (-4,-9), (-3,-8), (-2,-7), (-1,-6), (0,-5), (1,-4), (2,-3), (3,-2), (4,-1), (5,0) \][/tex]
2. Vertex:
The function [tex]\( f(x) = x - 5 \)[/tex] is a linear function, meaning it represents a straight line. Linear functions do not have vertices or maximum/minimum points. Hence:
- Vertex: None
- Maximum/Minimum point: None
3. Axis of Symmetry:
The axis of symmetry is a vertical line that passes through the vertex for parabola functions or higher-order polynomials. Since [tex]\( f(x) = x - 5 \)[/tex] is a straight line and does not form a parabola, it does not have an axis of symmetry. Therefore:
- Axis of Symmetry: None
4. Transformation Description:
The function [tex]\( f(x) = x - 5 \)[/tex] can be understood as a transformation of the basic linear function [tex]\( f(x) = x \)[/tex]. Specifically, it represents a vertical shift of the function [tex]\( f(x) = x \)[/tex] downward by 5 units.
Summary:
1. Points: [tex]\[ (-5,-10), (-4,-9), (-3,-8), (-2,-7), (-1,-6), (0,-5), (1,-4), (2,-3), (3,-2), (4,-1), (5,0) \][/tex]
2. Vertex: None
3. Maximum/Minimum: None
4. Axis of Symmetry: None
5. Transformation: The function [tex]\( f(x) = x - 5 \)[/tex] represents a downward shift of 5 units of the function [tex]\( f(x) = x \)[/tex].
Graph:
On a Cartesian plane, plot the points calculated above and draw a straight line through them to sketch the graph accurately. The line will pass through points such as [tex]\((-5, -10)\)[/tex], [tex]\((0, -5)\)[/tex], and [tex]\((5, 0)\)[/tex].
