Question

use the graph of the function to complete the table.
input | output
2 |
4 | 7
6 | 5

Explanation:

Step1: Locate x=2 on the graph

Find the point on the x - axis where x = 2. Then, move up (or down) to the graph of the function.

Step2: Determine the y - value at x=2

From the graph, when x = 2, we look at the corresponding y - coordinate. By visually inspecting the graph, at x = 2, the y - value (output) is 5? Wait, no, wait. Wait, let's re - check. Wait, the graph: when x = 2, let's see the curve. Wait, the grid: each square is 1 unit. At x = 2, the graph is at y = 5? Wait, no, wait, when x=0, y=0; x=2, let's see the line. Wait, maybe I made a mistake. Wait, the input x = 2, we need to find the output (y - value) from the graph. Let's look at the graph again. The curve passes through x = 2, what's the y? Wait, the table has x=4, y=7; x=6, y=5. For x=2, let's see the graph. At x = 2, the y - coordinate is 5? Wait, no, wait, maybe it's 5? Wait, no, let's count the grid. The x - axis: 0,1,2,3,4,5,6,7,8,9,10. The y - axis: 0,1,2,3,4,5,6,7,8,9,10. At x = 2, the graph is at y = 5? Wait, no, when x=2, the curve is at y = 5? Wait, no, maybe I miscalculated. Wait, let's see the slope. From x=0 (y=0) to x=4 (y=7). Wait, no, at x=2, let's see the graph. Wait, the user's graph: the curve starts at (0,0), goes up to (4,7), then down to (8,0). So at x=2, what's the y? Let's use symmetry? Wait, the function is a parabola? Wait, the graph is a parabola opening downwards with vertex at (4,7), roots at (0,0) and (8,0). So the equation of the parabola can be written as \(y=a(x - 0)(x - 8)=ax(x - 8)\). When x = 4, y=7. So \(7=a\times4\times(4 - 8)=a\times4\times(- 4)=-16a\), so \(a=-\frac{7}{16}\). Then the equation is \(y =-\frac{7}{16}x(x - 8)\). When x = 2, \(y=-\frac{7}{16}\times2\times(2 - 8)=-\frac{7}{16}\times2\times(-6)=\frac{7\times12}{16}=\frac{21}{4}=5.25\)? Wait, but maybe the graph is drawn with integer values. Wait, maybe the table's x=2, the output is 5? Wait, no, maybe I made a mistake. Wait, the user's table: input 2, output? Let's look at the graph again. The graph at x=2: looking at the grid, each square is 1 unit. At x=2, the y - coordinate is 5? Wait, no, when x=2, the graph is at y = 5? Wait, maybe the answer is 5? Wait, no, let's check the graph again. Wait, the curve at x=2: if we look at the graph, when x=2, the y - value is 5? Wait, maybe I was wrong earlier. Wait, the user's graph: the table has x=4, y=7; x=6, y=5. So at x=2, what's the y? Let's see the pattern. From x=0 (y=0) to x=4 (y=7): the increase is 7 over 4 units. From x=4 (y=7) to x=8 (y=0): decrease of 7 over 4 units. So at x=2 (which is 2 units from x=0), the y - value should be 5? Wait, no, 0 to 4: 4 units, y from 0 to 7. So per unit x, the increase is 7/4 per unit? No, that's not linear. Wait, it's a parabola. So using the equation \(y =-\frac{7}{16}x(x - 8)\). For x=2: \(y=-\frac{7}{16}\times2\times(-6)=\frac{84}{16}=\frac{21}{4}=5.25\), but maybe the graph is drawn with integer values, so maybe 5? Wait, no, maybe the answer is 5? Wait, no, let's check the graph again. The user's graph: at x=2, the y - coordinate is 5? Wait, maybe I made a mistake. Wait, the correct way is to look at the graph. So when x=2, we find the point on the graph with x - coordinate 2, then read the y - coordinate. From the graph, at x=2, the y - value is 5? Wait, no, maybe it's 5. Wait, the table has x=4, y=7; x=6, y=5. So x=2: let's see the symmetry. The parabola is symmetric about x=4. So the distance from x=2 to x=4 is 2, and the distance from x=6 to x=4 is also 2. So the y - value at x=2 should be equal to the y - value at x=6? But x=6 has y=5, so x=2 should also have y=5? Wait, that's the symmetry of the parabola (since the axis of symmetry is x=4). So yes, because the axis of symmetry is x = 4, so f(2)=f(6). Since f(6)=5, then f(2)=5.

Answer:

5