3 Graphing Quadratic Functions Worksheet - Free Printable

You’ve uploaded a worksheet titled “Graphing Quadratic Functions” with four problems. Each problem gives a quadratic function in vertex form (or standard form), and asks you to find:

- Vertex
- y-intercept
- x-intercept(s)

Then, you’re expected to graph the parabola on the provided coordinate grid.

Let’s solve each one step-by-step.

---

## 🔹 Problem 1: \( f(x) = x^2 \)

This is the parent quadratic function.

✔ Vertex:

Standard form: \( f(x) = a(x - h)^2 + k \) → here, \( h = 0, k = 0 \)
→ Vertex = (0, 0)

✔ y-intercept:

Set \( x = 0 \):
\( f(0) = 0^2 = 0 \)
→ y-intercept = (0, 0)

✔ x-intercept(s):

Set \( f(x) = 0 \):
\( x^2 = 0 \) → \( x = 0 \)
→ x-intercept = (0, 0)

> So this parabola touches the origin and opens upward.

---

## 🔹 Problem 2: \( f(x) = x^2 + 5 \)

This is a vertical shift of the parent function up by 5 units.

✔ Vertex:

Rewrite as \( f(x) = (x - 0)^2 + 5 \) → \( h = 0, k = 5 \)
→ Vertex = (0, 5)

✔ y-intercept:

Set \( x = 0 \):
\( f(0) = 0^2 + 5 = 5 \)
→ y-intercept = (0, 5)

✔ x-intercept(s):

Set \( f(x) = 0 \):
\( x^2 + 5 = 0 \) → \( x^2 = -5 \) → No real solution
→ No x-intercepts

> Parabola opens upward, vertex at (0,5), never crosses x-axis.

---

## 🔹 Problem 3: \( f(x) = (x + 3)^2 \)

This is a horizontal shift of the parent function left by 3 units.

✔ Vertex:

Rewrite as \( f(x) = (x - (-3))^2 + 0 \) → \( h = -3, k = 0 \)
→ Vertex = (-3, 0)

✔ y-intercept:

Set \( x = 0 \):
\( f(0) = (0 + 3)^2 = 9 \)
→ y-intercept = (0, 9)

✔ x-intercept(s):

Set \( f(x) = 0 \):
\( (x + 3)^2 = 0 \) → \( x = -3 \)
→ x-intercept = (-3, 0)

> Parabola opens upward, touches x-axis at vertex (-3,0).

---

## 🔹 Problem 4: \( f(x) = (x - 4)^2 - 3 \)

This is shifted right 4 units and down 3 units.

✔ Vertex:

Already in vertex form: \( f(x) = (x - 4)^2 - 3 \) → \( h = 4, k = -3 \)
→ Vertex = (4, -3)

✔ y-intercept:

Set \( x = 0 \):
\( f(0) = (0 - 4)^2 - 3 = 16 - 3 = 13 \)
→ y-intercept = (0, 13)

✔ x-intercept(s):

Set \( f(x) = 0 \):
\( (x - 4)^2 - 3 = 0 \)
→ \( (x - 4)^2 = 3 \)
→ \( x - 4 = \pm\sqrt{3} \)
→ \( x = 4 \pm \sqrt{3} \)

Approximate values:
\( \sqrt{3} \approx 1.732 \)
→ \( x \approx 4 + 1.732 = 5.732 \)
→ \( x \approx 4 - 1.732 = 2.268 \)

→ x-intercepts ≈ (2.27, 0) and (5.73, 0)

> Parabola opens upward, vertex at (4,-3), crosses x-axis at two points.

---

## 📊 Summary Table:

| Function | Vertex | y-intercept | x-intercept(s) |
|----------------------|------------|-------------|--------------------------------|
| 1. \( f(x) = x^2 \) | (0, 0) | (0, 0) | (0, 0) |
| 2. \( f(x) = x^2 + 5 \) | (0, 5) | (0, 5) | None |
| 3. \( f(x) = (x + 3)^2 \) | (-3, 0) | (0, 9) | (-3, 0) |
| 4. \( f(x) = (x - 4)^2 - 3 \) | (4, -3) | (0, 13) | \( 4 \pm \sqrt{3} \) ≈ (2.27, 0), (5.73, 0) |

---

## 🖍️ Graphing Tips:

- All these are upward-opening parabolas (since coefficient of \( x^2 \) is positive).
- Plot the vertex first.
- Plot the y-intercept.
- If there are x-intercepts, plot those too.
- Use symmetry: for example, in #4, since vertex is at x=4, and one x-intercept is at ~2.27, the other should be symmetric: 4 + (4 - 2.27) = 5.73 — which matches!
- Sketch a smooth U-shaped curve through the points.

---

✔ You can now fill in the blanks on your worksheet and sketch each graph accordingly.

Let me know if you’d like me to generate actual graphs or printable answer sheets!