Question

use transformations of $f(x)=x^2$ to graph the following function. $g(x)= -3(x - 1)^2 - 3$ use the graphing tool to graph the function. click to enlarge graph

Explanation:

Step1: Analyze Vertical Stretch/Reflection

The function \( g(x) = -3(x - 1)^2 - 3 \) has a coefficient of \(-3\) for the squared term. Compared to \( f(x)=x^2 \), the \(-3\) means we first vertically stretch \( f(x) \) by a factor of \( 3 \) (since \( |3| = 3 \)) and then reflect it over the \( x \)-axis (because of the negative sign). So the graph of \( f(x) \) becomes narrower and opens downward after this transformation.

Step2: Analyze Horizontal Shift

The term \( (x - 1) \) inside the square means we shift the graph of the function (after the vertical transformation) \( 1 \) unit to the right. For a function \( f(x - h) \), the graph shifts \( h \) units to the right when \( h>0 \). Here, \( h = 1 \), so shift right by \( 1 \).

Step3: Analyze Vertical Shift

The \(-3\) at the end of the function means we shift the graph (after horizontal shift) \( 3 \) units down. For a function \( f(x)+k \), when \( k<0 \), it shifts down by \( |k| \) units. Here, \( k=-3 \), so shift down by \( 3 \) units.

Step4: Identify Vertex

The vertex of \( f(x)=x^2 \) is at \( (0,0) \). After the transformations: horizontal shift right \( 1 \), vertical shift down \( 3 \), and reflection/stretch, the vertex of \( g(x) \) is at \( (1, -3) \). The parabola opens downward (due to the negative coefficient) and is narrower (due to \( |3|>1 \)).

To graph, start with the vertex \( (1, -3) \), then use the vertical stretch (e.g., when \( x = 2 \), \( g(2)=-3(2 - 1)^2 - 3=-3(1)-3=-6 \); when \( x = 0 \), \( g(0)=-3(0 - 1)^2 - 3=-3(1)-3=-6 \)) to plot other points and draw the parabola opening downward with vertex at \( (1, -3) \), narrower than \( f(x)=x^2 \), shifted right \( 1 \) and down \( 3 \).

Answer:

To graph \( g(x)=-3(x - 1)^2 - 3 \) from \( f(x)=x^2 \):

1. Vertically stretch \( f(x) \) by factor \( 3 \), reflect over \( x \)-axis (opens downward, narrower).
2. Shift the result \( 1 \) unit right (vertex moves to \( x = 1 \)).
3. Shift \( 3 \) units down (vertex moves to \( y=-3 \)).

The vertex is \( (1, -3) \), parabola opens downward, narrower than \( x^2 \), shifted right \( 1 \), down \( 3 \). (Use these transformations to plot points and draw the graph.)