This transformation involves subtracting a constant value (k = -10) from the original function M(x). This results in a **vertical shift** of the graph. Since we are subtracting, the graph will shift **downwards** by 10 units.
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To visualize this, consider each point (x, y) on the original graph of M(x). The corresponding point on the transformed graph will be (x, y - 10). This means that every y-coordinate is reduced by 10, effectively shifting the entire graph downwards.
M(x - 10): Horizontal Translation
In this transformation, a constant (k = -10) is subtracted from the input variable x inside the function. This causes a **horizontal shift** of the graph. Since we are subtracting from x, the graph shifts to the **right** by 10 units.
To understand this, think about the input values. For the original function M(x), an input of x = 5 would produce a certain output. Now, in the transformed function M(x - 10), we need an input of x = 15 (because 15 - 10 = 5) to get the same output. This shows that the entire graph has moved 10 units to the right.
-10M(x): Vertical Stretch or Compression
This transformation involves multiplying the original function M(x) by a constant factor (k = -10). This leads to a **vertical stretch or compression** of the graph, depending on the absolute value of k.
In this case, k = -10, which is a negative number. This means the graph will undergo a **vertical stretch by a factor of 10** and also a **reflection about the x-axis**.
For each point (x, y) on the original graph, the corresponding point on the transformed graph will be (x, -10y). The y-coordinate is multiplied by -10, making the graph taller (stretched) and flipping it across the x-axis.
M(-10x): Horizontal Stretch or Compression
Here, the input variable x is multiplied by a constant (k = -10) within the function. This results in a **horizontal stretch or compression** of the graph. Similar to the vertical case, the direction of the stretch or compression depends on the absolute value of k.
Since k = -10, the graph will undergo a **horizontal compression by a factor of 1/10** and also a **reflection about the y-axis**.
Imagine a point (x, y) on the original graph. On the transformed graph, the corresponding point becomes (-x/10, y). This means the x-coordinate is divided by -10, making the graph narrower (compressed) and reflecting it across the y-axis.