Let’s analyze the graph and answer the questions step by step.

(c) Axis of Symmetry and Height Behavior

• Axis of Symmetry: The graph shows a vertical parabola that is symmetric around a line. The axis of symmetry in a parabola is a vertical line that passes through the vertex. The vertex is the highest point of the graph (since it opens downward).

  From the graph, the axis of symmetry is the vertical line passing through $t = 0$. This line divides the parabola into two mirror images.

  Height Behavior:

  • For $t < 0$, the height is increasing.
  • For $t > 0$, the height is decreasing.

  This is typical of a parabolic motion where an object is thrown upwards: it rises to a maximum height (vertex) and then starts descending after that.

(d) Intercepts

• t-intercepts (x-intercepts): These are the points where the graph crosses the horizontal axis (the time axis).

  • From the graph, the t-intercepts appear to be $t = -a$ and $t = a$, where $a$ is approximately a small value near 0 (likely $t = -2$ and $t = 2$). These represent the times when the object touches the ground (height is zero).

• h-intercept (y-intercept): This is the point where the graph crosses the vertical axis (the height axis).

  • The graph crosses the h-axis at approximately $h = 200$, which represents the maximum height reached by the object at $t = 0$.

Interpretation:

• t-intercepts: These represent the time moments when the object is at ground level (launch and return).
• h-intercept: This represents the maximum height the object reaches when $t = 0$.

Questions:

1. How do you determine the vertex of a parabola analytically if given a quadratic equation?
2. What would happen to the axis of symmetry if the graph were shifted horizontally?
3. Can you explain why the t-intercepts are symmetrically placed around the axis of symmetry?
4. How does the width of the parabola relate to the speed of the object in this scenario?
5. If the maximum height were different, how would the vertex and the intercepts shift?

Tip:

For parabolas, the equation $h = at^2 + bt + c$ provides direct information about the intercepts and the axis of symmetry based on the coefficients.