Question

use the graph of the function to complete the table.

input | output
5 | 3
6 |
7 |

Explanation:

Step1: Analyze the graph at x=6

The graph is a parabola. At x = 6, it's the vertex (lowest point). By observing the grid, the y - value (output) at x = 6 is 1 (since it's halfway between 0 and 2 on the y - axis, and the grid lines are at intervals of 1? Wait, no, looking at the y - axis: from 0 to 2 is one interval, 2 to 4 another, etc. Wait, the vertex is at (6,1)? Wait, no, let's check the coordinates. The graph at x = 6: the point is at y = 1? Wait, no, maybe the grid has each square as 1 unit. Let's see, the y - axis has marks at 0,2,4,6,8,10. The vertex is at (6,1)? Wait, no, maybe I misread. Wait, the input is 6, so we look at x = 6 on the graph. The graph's lowest point (vertex) is at x = 6, and the y - coordinate there: looking at the grid, each small square is 1 unit. So at x = 6, the y - value is 1? Wait, no, maybe the y - axis is labeled with 0,2,4,6,8,10, but the space between 0 and 2 is two small squares? Wait, no, the graph: when x = 4, y = 10; x = 8, y = 10; x = 6, the minimum. Let's see the table: input 5, output 3. So at x = 5, y = 3. Then x = 6: the vertex. Let's check the symmetry. The parabola is symmetric about x = 6 (since x = 4 and x = 8 have the same y - value, 10? Wait, no, x = 4: y = 10, x = 8: y = 10. So the axis of symmetry is x=(4 + 8)/2=6. So for x = 5 and x = 7 (since 5 is 1 unit left of 6, 7 is 1 unit right of 6), their y - values should be equal. Input 5 has output 3, so input 7 should also have output 3. And input 6: the vertex, so the minimum y - value. Let's see the graph: at x = 6, the y - value is 1? Wait, no, maybe the y - axis is such that each grid line is 1 unit. Wait, the graph at x = 6: the point is at y = 1? Wait, no, maybe I made a mistake. Wait, the table has input 5, output 3. So when x = 5, y = 3. Then x = 6: let's look at the graph. The vertex is at (6,1)? Wait, no, the distance from x = 5 to x = 6 is 1, and from x = 6 to x = 7 is 1. Since the parabola is symmetric about x = 6, f(5)=f(7)=3, and f(6) is the minimum. Let's check the graph: at x = 6, the y - coordinate is 1? Wait, no, maybe the y - axis is labeled with 0,2,4,6,8,10, but the actual values: when x = 4, y = 10; x = 8, y = 10; x = 5, y = 3; so x = 6, what's y? Let's see the grid. The vertical axis (y) has marks at 0,2,4,6,8,10. The horizontal axis (x) has marks at 0,2,4,6,8,10. The graph at x = 6: the point is at y = 1? Wait, no, maybe the y - value at x = 6 is 1. Wait, but let's think again. The problem is to find the output for input 6 and 7.

Step2: Find output for x = 6

Looking at the graph, at x = 6 (input = 6), the y - value (output) is 1 (since it's the vertex, the lowest point).

Step3: Find output for x = 7

Due to the symmetry of the parabola (axis of symmetry x = 6), the function values at x = 5 and x = 7 are equal. Since input 5 has output 3, input 7 will also have output 3.

Answer:

For input 6, output is 1; for input 7, output is 3. (Assuming the table has two blanks: one for 6, one for 7. The first blank (input 6) is 1, the second (input 7) is 3.)