Question

evaluate independent practice learning goals ✓ i can translate graphs of absolute value functions. (explain 1) ✓ i can stretch, shrink, and reflect graphs of absolute value functions. (explain 2) lesson reflection (circle one) starting... getting there... got it! starting... getting there... got it! lesson 1.1 checkpoint □ complete the previous problems, check your solutions, then complete the lesson checkpoint below. □ complete the lesson reflection above by circling your current understanding of the learning goals. describe the transformations from the graph of ( f(x) = |x| ) to the graph of the given function. then graph the given function. (see example 4) 36. ( v(x) = -3|x + 1| + 4 ) error analysis describe and correct the error in graphing the function. 46. ( y = -3|x| )

Explanation:

Problem 36: Transformations of \( v(x) = -3|x + 1| + 4 \) from \( f(x) = |x| \)

Step 1: Recall Transformations of Absolute Value Functions

The general form of an absolute value function is \( g(x) = a|x - h| + k \), where:

• \( a \): vertical stretch/shrink and reflection (if \( a < 0 \), reflect over x - axis; \( |a| > 1 \) stretch, \( 0 < |a| < 1 \) shrink)
• \( h \): horizontal shift (shift right if \( h > 0 \), left if \( h < 0 \))
• \( k \): vertical shift (shift up if \( k > 0 \), down if \( k < 0 \))

For \( f(x)=|x| \) and \( v(x)=-3|x + 1|+4 \):

Step 2: Analyze Horizontal Shift (\( h \))

In \( v(x)=-3|x - (- 1)|+4 \), \( h=-1 \). So, the graph of \( f(x) \) is shifted left by 1 unit.

Step 3: Analyze Vertical Stretch/Reflection (\( a \))

Here, \( a = - 3 \). Since \( |a|=3>1 \), the graph is vertically stretched by a factor of 3. Also, since \( a=-3<0 \), the graph is reflected over the x - axis.

Step 4: Analyze Vertical Shift (\( k \))

Since \( k = 4>0 \), the graph is shifted up by 4 units.

Step 1: Recall the Effect of \( a \) in \( y = a|x| \)

For the function \( y=-3|x| \), the coefficient \( a=-3 \). When \( a < 0 \) in the absolute value function \( y = a|x| \), the graph of \( y = |x| \) (which opens upwards) should be reflected over the \( x \) - axis (so it opens downwards) and vertically stretched by a factor of \( |a| = 3 \).

Step 2: Identify the Error

The given graph of \( y=-3|x| \) opens upwards, but the correct graph of \( y=-3|x| \) should open downwards (because of the negative sign in front of the \( 3|x| \)) after being vertically stretched by a factor of 3. The error is that the graph was not reflected over the \( x \) - axis; it still opens upwards like \( y = |x| \) instead of opening downwards as it should for \( y=-3|x| \).

Step 3: Correct the Graph

To correct the graph:

• Reflect the graph over the \( x \) - axis (so that it opens downwards).
• Ensure the vertical stretch by a factor of 3 is applied (the slope of the sides should be \( \pm3 \), but the direction should be downwards from the vertex at \( (0,0) \)).

Answer:

1. Reflect the graph of \( f(x)=|x| \) over the \( x \) - axis.
2. Vertically stretch the graph by a factor of 3.
3. Shift the graph 1 unit to the left.
4. Shift the graph 4 units up.

Problem 46: Error Analysis of \( y=-3|x| \) Graph