Question

graph the function and its parent function. then describe the transformation.

1. $f(x) = 5x$
2. $f(x) = 3(x + 2)^2$
3. $f(x) = -|x + 4| - 3$

(images of coordinate grids for each function are present, with the first having a graphed line and the note vertical stretch by 5)

Explanation:

Problem 5: \( f(x) = 5x \)

Step 1: Identify Parent Function

The parent function of a linear function \( f(x) = mx + b \) (where \( b = 0 \) here) is \( y = x \), which is a linear function with slope \( 1 \) and passes through the origin.

Step 2: Analyze Transformation

For the function \( f(x) = 5x \), we compare it to the parent function \( y = x \). The general form of a vertical stretch or compression is \( y = a \cdot g(x) \), where \( g(x) \) is the parent function. If \( |a| > 1 \), it is a vertical stretch by a factor of \( |a| \). Here, \( a = 5 \) and \( g(x)=x \), so the transformation is a vertical stretch of the parent function \( y = x \) by a factor of \( 5 \).

Step 1: Identify Parent Function

The parent function of a quadratic function in the form \( f(x)=a(x - h)^2 + k \) is \( y = x^2 \), which is a parabola opening upwards with vertex at the origin \((0,0)\).

Step 2: Analyze Horizontal Shift

For the horizontal shift, the form is \( y=(x - h)^2 \), where \( h \) determines the horizontal shift. If \( h<0 \), the shift is to the left. In \( f(x)=3(x + 2)^2 \), we can rewrite it as \( f(x)=3(x-(- 2))^2 \), so \( h=-2 \), which means the graph is shifted 2 units to the left from the parent function \( y = x^2 \).

Step 3: Analyze Vertical Stretch

The coefficient \( a = 3 \) in \( f(x)=3(x + 2)^2 \). For the vertical stretch or compression, if \( |a|>1 \), it is a vertical stretch by a factor of \( |a| \). So here, there is a vertical stretch by a factor of \( 3 \) in addition to the horizontal shift.

Step 1: Identify Parent Function

The parent function of an absolute - value function in the form \( f(x)=a|x - h|+k \) is \( y = |x| \), which is a V - shaped graph with vertex at the origin \((0,0)\) and opening upwards.

Step 2: Analyze Horizontal Shift

For the horizontal shift, we rewrite \( f(x)=-|x + 4|-3 \) as \( f(x)=-|x-(-4)|-3 \). Using the form \( y = |x - h| \), when \( h=-4 \), the graph is shifted 4 units to the left from the parent function \( y = |x| \).

Step 3: Analyze Reflection

The coefficient \( a=-1 \) in front of the absolute - value. If \( a=-1 \), the graph of the parent function \( y = |x| \) (which opens upwards) is reflected over the \( x \) - axis.

Step 4: Analyze Vertical Shift

The constant term \( k=-3 \) in the form \( y = a|x - h|+k \). A negative \( k \) value means the graph is shifted downwards. So the graph is shifted 3 units down from the reflected and horizontally shifted graph.

Answer:

The parent function is \( y = x \). The transformation of \( f(x)=5x \) from its parent function \( y = x \) is a vertical stretch by a factor of \( 5 \).

Problem 6: \( f(x) = 3(x + 2)^2 \)