The graph is upward u shape with points (3,-9), (1,-5),(5,-5),(0,0)(5,0)(-1,7),(7,7)

Based on the given points and the shape of the graph being an upward U-shape, we can analyze the information provided.

1. The vertex: Since we have points that appear to be the lowest on the left and then rise on the right, we need to find the minimum point of the U shape. Since the lowest point (vertex) is not explicitly given in the points you've provided, we interpolate based on the provided points. Given the points you provided, a good estimate for the vertex would be between (1,-5) and (3,-9). More specifically, the vertex seems to be at (3, -9).

2. The lesser zero: Zeros occur where the graph intersects the x-axis (f(x) = 0). From the points listed, we can see there are zeros at x = 0 and x = 5. The lesser zero is thus 0.

3. The other zero: As mentioned above, the other zero would be at 5.

4. The line of symmetry: The line of symmetry in a parabolic graph is a vertical line that passes through the vertex. Thus, the line of symmetry is x = 3.

5. The y-intercept: The y-intercept is the value of f(x) when x = 0. From the points provided, we see that f(0) = 0.

Now, we can complete the f(x) table for the specific x values:

| x | f(x) | |----|-------| | -1 | 7 | (Given point (−1, 7)) | | 0 | 0 | (From y-intercept) | | 1 | -5 | (Given point (1, -5)) | | 3 | -9 | (Vertex (3, -9)) | | 5 | 0 | (Given point (5, 0)) |

Summary of Answers:

• The vertex is (3, -9).
• The lesser zero is 0.
• The other zero is 5.
• The line of symmetry is x = 3.
• The y-intercept is 0.

The completed function table is as follows.

| x | f(x) | |----|-------| | -1 | 7 | | 0 | 0 | | 1 | -5 | | 3 | -9 | | 5 | 0 |

No problem! Let's revise the information based on the new point (6, 0) instead of (5, 0).

1. The vertex remains the same: (3, -9).

2. The lesser zero is still 0.

3. The other zero is now updated to 6 since the point (6, 0) indicates the x-intercept.

4. The line of symmetry remains x = 3.

5. The y-intercept still remains 0.

Now, let's complete the f(x) table accurately with the updated point.

| x | f(x) | |----|-------| | -1 | 7 | (Given point (−1, 7)) | | 0 | 0 | (From y-intercept) | | 1 | -5 | (Given point (1, -5)) | | 3 | -9 | (Vertex (3, -9)) | | 6 | 0 | (New point (6, 0)) |

Summary of Updated Answers:

• The vertex is (3, -9).
• The lesser zero is 0.
• The other zero is 6.
• The line of symmetry is x = 3.
• The y-intercept is 0.

The completed table is:

| x | f(x) | |----|-------| | -1 | 7 | | 0 | 0 | | 1 | -5 | | 3 | -9 | | 6 | 0 |

Let me know if you need any further adjustments!

Thank you for the clarification! Since the last x-value in your original table should be 5, and with the new point (6, 0) replacing the old point (5, 0), we should find the value of \( f(5) \).

From the context of the provided points, we need to determine \( f(5) \). Since the graph is upward and the vertex is at (3, -9), we know:

• The points around the vertex indicate that when x = 5, the f(x) value would be above the vertex. As such, we can refer to the points we do know:
  • The graph is rising as it approaches 5 from vertex (3, -9).
  • The value at (\( 6, 0 \)) tells us that when x = 6, \( f(6) = 0 \).

To approximate \( f(5) \):

• We can reasonably assume that since the parabola is symmetric around the vertex (3, -9), and since it approaches the x-axis around x = 6 (which is at 0), the \( f(5) \) point will be higher than \( f(6) = 0 \).

Given the symmetry and smoothness of parabola, a rough estimate would suggest that:

• Since (6, 0) is where the curve touches the x-axis, and (5, 0) would also typically approach close to zero, based on the nature of parabolas, we can infer that well above this point but still remaining below zero will yield \( f(5) \) to also be close to 0.

However, without additional calculation or more specific data points, we can only approximate.

Final point estimation: The approximate value for \( f(5) \) would thus be about -4 (since it must be below the x-axis before returning to zero at x = 6).

Thus, here’s the completed table again:

| x | f(x) | |----|-------| | -1 | 7 | | 0 | 0 | | 1 | -5 | | 3 | -9 | | 5 | -4 | * (estimate)* | | 6 | 0 |

If you have more specific data points or if there's more clarity on how to arrive at \( f(5) \), please share!