This worksheet helps students visualize how changing the constants a, h, and k affects the graph of a quadratic function in vertex form.

It looks like you've uploaded a worksheet titled "Transformations of Quadratic Functions" that asks students to analyze how the constants $ a $, $ h $, and $ k $ in the quadratic function $ f(x) = a(x - h)^2 + k $ affect the graph of the parabola.

Let’s go through the table and fill in the missing information based on standard transformations of quadratic functions. The goal is to understand how each constant affects the graph.

---

🔷 General Form:

$$
f(x) = a(x - h)^2 + k
$$

This is the vertex form of a quadratic function, where:
- $ (h, k) $ is the vertex of the parabola,
- $ a $ determines the direction and width of the parabola,
- $ h $ controls horizontal shifts,
- $ k $ controls vertical shifts.

---

✔ Let's complete the table:

| Constant | Examples (graph these...) | Visual Representation | Describe what changed... |
|---------|----------------------------|------------------------|----------------------------|
| k | $ f(x) = x^2 $<br> $ g(x) = x^2 + 5 $<br> $ h(x) = x^2 - 3 $<br> $ j(x) = x^2 + 2 $ | [Graphs: up/down shifts] | Changing the value of $ k $ shifts the graph vertically.<br> If $ k $ is positive → graph shifts up.<br> If $ k $ is negative → graph shifts down. |
| h | $ f(x) = x^2 $<br> $ g(x) = (x - 2)^2 $<br> $ h(x) = (x + 5)^2 $<br> $ j(x) = (x - 1)^2 $<br> $ k(x) = (x + 4)^2 $ | [Graphs: left/right shifts] | Changing the value of $ h $ shifts the graph horizontally.<br> If $ h $ is positive, the graph shifts right.<br> If $ h $ is negative, the graph shifts left. *(Note: It's $ x - h $, so if $ h = 2 $, it's $ x - 2 $, which shifts right by 2.)* |
| a | $ f(x) = x^2 $<br> $ g(x) = 3x^2 $<br> $ h(x) = \frac{1}{2}x^2 $<br> $ j(x) = -x^2 $<br> $ k(x) = -\frac{1}{4}x^2 $ | [Graphs: stretch/compress/invert] | Changing the value of $ a $ stretches or compresses the graph.<br> If $ |a| > 1 $, the graph narrows.<br> If $ 0 < |a| < 1 $, the graph widens.<br> If $ a < 0 $, the graph opens downward (reflected over the x-axis). |
| ± | $ f(x) = x^2 $<br> $ g(x) = -x^2 $ | [Graphs: reflection] | A negative sign in front of the equation (i.e., $ a < 0 $) reflects the graph over the x-axis. |

---

📌 Key Notes:

- The sign of $ a $ determines the direction of the parabola:
- $ a > 0 $: opens up
- $ a < 0 $: opens down
- The magnitude of $ a $ determines how wide or narrow the parabola is.
- $ h $ and $ k $ shift the vertex from $ (0, 0) $ to $ (h, k) $.

---

✔ Final Answer Summary:

| Constant | Effect |
|--------|--------|
| k | Vertical shift: $ +k $ → up, $ -k $ → down |
| h | Horizontal shift: $ +h $ → right, $ -h $ → left (remember: $ x - h $) |
| a | Controls width and direction: $ |a| > 1 $ → narrower, $ |a| < 1 $ → wider; $ a < 0 $ → flips over x-axis |
| ± | Negative sign flips the graph over the x-axis |

---

Would you like me to generate a visual representation (e.g., sketch descriptions) for any of these transformations? Or help with graphing specific examples?