Question

a table of values for $y = \frac{1}{4}(x - 2)^2 + 8$ is shown, what features can be found using the table of values?□ x-intercepts□ y-intercepts□ vertex□ axis of symmetry□ increasing/decreasing□ positive/negative| x | y || ---- | ---- || -3 | 14.25 || -2 | 12 || -1 | 10.25 || 0 | 9 || 1 | 8.25 || 2 | 8 || 3 | 8.25 |

Explanation:

Step1: Identify vertex from table

Look for the minimum y-value: when $x=2$, $y=8$ (lowest in table).
Vertex: $(2, 8)$

Step2: Find axis of symmetry

Axis of symmetry is vertical line through vertex x-coordinate: $x=2$

Step3: Determine positive/negative nature

All y-values in table are positive, and the coefficient $\frac{1}{4}>0$, so the parabola opens upward (positive).

Step4: Find increasing/decreasing intervals

For $x<2$, y decreases as x increases (e.g., $x=-3,y=14.25$; $x=2,y=8$). For $x>2$, y increases as x increases (e.g., $x=2,y=8$; $x=3,y=8.25$).
Decreasing: $(-\infty, 2)$; Increasing: $(2, \infty)$

Step5: Check y-intercept

Y-intercept occurs at $x=0$. From table, $x=0,y=9$.
Y-intercept: $(0, 9)$

Step6: Check x-intercepts

All y-values in table are positive, and the minimum y-value is 8 (above x-axis). No x-intercepts exist.

Answer:

1. x-intercepts: None (no x-values make $y=0$)
2. y-intercepts: $(0, 9)$
3. vertex: $(2, 8)$
4. axis of symmetry: $x=2$
5. increasing/decreasing: Decreasing on $(-\infty, 2)$, Increasing on $(2, \infty)$
6. positive/negative: Positive (parabola opens upward, all y-values positive)