Problem 54 - Algebra and Trigonometry

Question

\(45-54=\) A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=|x|\) shrink vertically by a factor of \(\frac{1}{2},\) shift to the left 1 unit, and shift upward 3 units

Step-by-Step Solution

Answer

The final equation is \(i(x) = \frac{1}{2}|x+1| + 3\).

Step 1: Vertical Shrink by a Factor of 1/2

To vertically shrink the graph of the function \(f(x) = |x|\) by a factor of \(\frac{1}{2}\), multiply the entire function by \(\frac{1}{2}\). This gives us the new function:\[g(x) = \frac{1}{2}|x|\].

Step 2: Shift to the Left 1 Unit

To shift the function \(g(x) = \frac{1}{2}|x|\) to the left by 1 unit, replace \(x\) with \(x+1\). Thus, the function becomes:\[h(x) = \frac{1}{2}|x+1|\].

Step 3: Shift Upward 3 Units

To shift the function \(h(x) = \frac{1}{2}|x+1|\) upward by 3 units, add 3 to the entire function. So the final equation becomes:\[i(x) = \frac{1}{2}|x+1| + 3\].

Key Concepts

Vertical Shrink

When we talk about a vertical shrink, we are referring to how the graph of a function is compressed towards the x-axis. Imagine you're squishing the graph down by a specific factor, in this case, by \(\frac{1}{2}\). This is different from scaling horizontally as it affects the y-values directly. For the function \(f(x) = |x|\), the vertical shrink is achieved by multiplying the function by the shrinking factor. This process makes every y-value half its original size, resulting in \(g(x) = \frac{1}{2}|x|\). By visualising this, you can see that the graph effectively becomes "shorter" but retains its basic shape.

Vertical transformations like this one are straightforward once you remember to apply the multiplying factor to the entire function.

Horizontal Shift

Horizontal shifts move a function left or right along the x-axis, but contrary to intuition, it involves changing the input variable, \(x\). To shift a graph horizontally, you adjust the variable inside the function. A leftward shift by 1 unit means replacing \(x\) with \(x + 1\) in the function.

After vertically shrinking \(f(x) = |x|\), we have \(g(x) = \frac{1}{2}|x|\). For a left shift of 1 unit, the function transforms to \(h(x) = \frac{1}{2}|x + 1|\).

• Positive inside the absolute value implies a left shift.
• Negative inside means a right shift.

This might feel a bit backward, but think of it like adjusting the "starting point" of the function’s graph.

Vertical Shift

After applying the vertical shrink and horizontal shift, our function is now \(h(x) = \frac{1}{2}|x + 1|\). A vertical shift is akin to moving the graph up or down along the y-axis.

For upward movement, you add a value to the entire function, hence the term "shift upward". Here, adding 3 will raise every point on the graph, resulting in the final version of the function: \(i(x) = \frac{1}{2}|x + 1| + 3\).

• Add a positive value to shift up.
• Subtract a positive to shift down.

The graph is elevated without altering its shape, simply elevating the y-intercepts and maintaining the overall form.

Absolute Value Function

The absolute value function, represented as \(f(x) = |x|\), is fundamental in math, creating a V-shaped graph centered at the origin (0,0). The function's key property is that it always produces non-negative outputs, resulting from it taking the "absolute" or positive distance of any input \(x\) from zero.

This graph features a sharp point called the vertex at \(x = 0\), where the direction changes.

• It's symmetrical around the y-axis, indicating even function properties.
• For positive \(x\), \(f(x) = x\).
• For negative \(x\), \(f(x) = -x\).

Understanding the basics of absolute value helps in quickly performing transformations like vertical shrinks, horizontal shifts, and vertical shifts as you've seen above.