How do you use the important points to sketch the graph of #f(x) = 1/3(x + 5)^2 - 1#?

Question

Julia Adams · Accepted Answer

See explanation

Set:#" "y=1/3(x+5)^2-1#

Compare the graph to the given equation.

Observe that
#" "x_("vertex")=(-1)xx5=-5 #
#" "y_("vertex")= -1#

To find the #x_("intercepts")# set y=0
To find the #y_("intercepts")# set x=0

Mark the points on the axis then draw free hand the best curve you can that passes through these points. Do this remembering that the axis of symmetry is at #x_("vertex")#

Solve

Eli Assouline · Answer

undefined To sketch the graph of $ f(x) = \frac{1}{3}(x + 5)^2 - 1 $, follow these steps:

1. Identify the vertex: The vertex of the parabola represented by the function $ f(x) $ is given by the coordinates $(-5, -1)$.

2. Determine the direction of opening: Since the coefficient of $ x^2 $ is positive ($ \frac{1}{3} $), the parabola opens upwards.

3. Find the y-intercept: Substitute $ x = 0 $ into the equation to find the y-intercept. $ f(0) = \frac{1}{3}(0 + 5)^2 - 1 = \frac{1}{3}(5)^2 - 1 = \frac{25}{3} - 1 = \frac{22}{3} $. So, the y-intercept is $ (0, \frac{22}{3}) $.

4. Plot additional points: Choose additional x-values, plug them into the equation, and calculate corresponding y-values to plot more points if needed.

5. Sketch the graph: Use the vertex, direction of opening, and plotted points to sketch the graph of the function.

6. Label the vertex and any other significant points on the graph.

7. Draw the parabola through the plotted points, ensuring it opens upwards and has its vertex at $(-5, -1)$.