I need step by step solution with final answer.
Sketch the graph of f(x)=(1)/(2)(x-4)^(2)-1

Question

VIPSolutions · Accepted Answer

Let's analyze and sketch the graph of the function step by step:

$$
f(x) = \frac{1}{2}(x-4)^2 - 1
$$

### **Step 1: Identify the Type of Function**

This is a **quadratic function** in vertex form:  
$$
f(x) = a(x-h)^2 + k
$$
where $ a = \frac{1}{2} $, $ h = 4 $, $ k = -1 $.

---

### **Step 2: Find the Vertex**

The vertex of the parabola is at $(h, k)$:

$$
\text{Vertex}: (4, -1)
$$

---

### **Step 3: Axis of Symmetry**

The axis of symmetry is the vertical line $ x = h $:

$$
x = 4
$$

---

### **Step 4: Direction and Width**

- Since $ a = \frac{1}{2} > 0 $, the parabola **opens upwards**.
- The value $ |a| = \frac{1}{2} < 1 $ means the parabola is **wider** than the standard $ y = x^2 $.

---

### **Step 5: Y-intercept**

Set $ x = 0 $:

$$
f(0) = \frac{1}{2}(0-4)^2 - 1 = \frac{1}{2}(16) - 1 = 8 - 1 = 7
$$
So, the y-intercept is at $ (0, 7) $.

---

### **Step 6: X-intercepts (if any)**

Set $ f(x) = 0 $:

$$
\frac{1}{2}(x-4)^2 - 1 = 0 \
\frac{1}{2}(x-4)^2 = 1 \
(x-4)^2 = 2 \
x-4 = \pm \sqrt{2} \
x = 4 \pm \sqrt{2}
$$

So the x-intercepts are at $ (4+\sqrt{2},\ 0) $ and $ (4-\sqrt{2},\ 0) $.

---

### **Step 7: Sketch Key Points**

- **Vertex:** $ (4, -1) $
- **Y-intercept:** $ (0, 7) $
- **X-intercepts:** $ (4+\sqrt{2}, 0) $ and $ (4-\sqrt{2}, 0) $  
  (numerically, $ \sqrt{2} \approx 1.41 $, so $ x \approx 5.41 $ and $ x \approx 2.59 $)

---

### **Step 8: Sketch**

Here's an ASCII sketch (for visualization):

```
   y
   ^
  8|       *
  7|*         *
  6|
  5|
  4|
  3|
  2|
  1|
  0|         o       o
 -1|            V
    +--------------------> x
        2.59   4   5.41
```
- V = vertex (4, -1)
- * = y-intercept (0,7)
- o = x-intercepts (2.59, 0) and (5.41, 0)

---

## **Summary of Solution**

**Key features:**
- **Vertex:** $ (4, -1) $
- **Axis of symmetry:** $ x = 4 $
- **Y-intercept:** $ (0, 7) $
- **X-intercepts:** $ (4+\sqrt{2}, 0) $, $ (4-\sqrt{2}, 0) $
- **Opens upward, wider than $ y = x^2 $**

**Graph:**
- A U-shaped parabola with vertex at (4, -1), opening upward, passing through (0, 7), and crossing the x-axis at about $ x = 2.59 $ and $ x = 5.41 $.

---

If you need a more exact plot, you can use graphing tools or software! If you need it plotted on Desmos or another graphing calculator, let me know!

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