How do you sketch the graph of #y=(x+3)^2# and describe the transformation?

Question

Eva Abramson · Accepted Answer

#"see explanation"# #"the graph of " y=(x+3)^2" is the graph of " y=x^2# #"translated 3 units to the left"# #"the graph of " y=x^2" is "# graph{x^2 [-10, 10, -5, 5]} #"the graph of " y=(x+3)^2# #" the point " (0,0)to(-3,0)# #"choosing some values of x"# #x=0toy=3^2=9to(0,9)larr" y-intercept"# #x=-5toy=2^2=4to(-5,4)# #"plot these points and draw a smooth curve through them"# #"for the graph of " y=(x+3)^2# graph{(x+3)^2 [-10, 10, -5, 5]} #color(blue)"In general"# #y=f(x+a)" is f(x) translated a units left "larr# #y=f(x-a)" is f(x) translated a units right "rarr#

Hazel Anglin · Answer

undefined To sketch the graph of $ y = (x+3)^2 $, follow these steps:

1. Identify the parent function: $ y = x^2 $, which is a basic quadratic function with its vertex at the origin.
2. Apply the transformation $ (x+3) $ to the parent function, shifting it three units to the left.
3. The vertex of the transformed function will be at the point $(-3, 0)$.
4. Plot several points on the graph by choosing values for $ x $ and calculating the corresponding $ y $ values.
5. Draw a smooth curve through the plotted points to represent the graph of the transformed function.

The transformation shifts the graph of the parent function three units to the left without changing its shape.

Nathan Axel · Answer

undefined To sketch the graph of $ y = (x + 3)^2 $, follow these steps:

1. Start with the basic graph of $ y = x^2 $, which is a parabola opening upwards, centered at the origin.

2. Apply a horizontal shift of 3 units to the left. This means you move each point on the graph 3 units to the left. So, if you had a point (a, b) on the original graph, it becomes (a - 3, b) on the new graph.

3. The resulting graph will be a parabola opening upwards, with its vertex at the point (-3, 0). It will be shifted 3 units to the left compared to the basic $ y = x^2 $ graph.

This transformation is a horizontal translation of the graph to the left by 3 units.