Question

graph (h(t) = 2(t + 4)^2 + 3).

Explanation:

⚡ Using what you learned: graphing quadratic functions

Step 1: Identify the vertex

The quadratic function is given in vertex form:

$$ h(t) = a(t - h)^2 + k $$

Comparing \( h(t) = 2(t + 4)^2 + 3 \) to the standard vertex form:

• \( a = 2 \)
• \( h = -4 \)
• \( k = 3 \)

The vertex is at \( (-4, 3) \).

Step 2: Find additional points

Evaluate \( h(t) \) for values of \( t \) near the vertex \( t = -4 \):

For \( t = -3 \):

$$ h(-3) = 2(-3 + 4)^2 + 3 = 2(1)^2 + 3 = 5 \implies (-3, 5) $$

For \( t = -5 \):

$$ h(-5) = 2(-5 + 4)^2 + 3 = 2(-1)^2 + 3 = 5 \implies (-5, 5) $$

For \( t = -2 \):

$$ h(-2) = 2(-2 + 4)^2 + 3 = 2(2)^2 + 3 = 11 \implies (-2, 11) $$

Step 3: Plot the points and sketch the parabola

1. Plot the vertex at \( (-4, 3) \).
2. Plot the symmetric points at \( (-3, 5) \) and \( (-5, 5) \).
3. Draw a smooth, upward-opening parabola passing through these points.

Answer:

To graph the function \( h(t) = 2(t+4)^2 + 3 \):

1. Vertex: Plot the lowest point of the parabola at \( (-4, 3) \).
2. Symmetric Points: Plot the points \( (-5, 5) \) and \( (-3, 5) \).
3. Curve: Draw a U-shaped curve (parabola) opening upwards through these plotted points.