Worksheet for graphing quadratic functions with a=1, featuring four equations and corresponding coordinate grids.

You're given a worksheet titled “Graph quadratic functions with a=1” — meaning all the parabolas open upward (since the leading coefficient is positive 1) and have the same “width” as the parent function $ y = x^2 $. Your task is to graph each quadratic function by identifying key features: vertex, axis of symmetry, y-intercept, and possibly x-intercepts or additional points.

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Let’s solve each one step-by-step and explain how to graph them.

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## 🔹 Problem 1: $ y = x^2 - 6x + 4 $

Step 1: Find the vertex

Use the formula for the x-coordinate of the vertex:
$ x = \frac{-b}{2a} $

Here, $ a = 1 $, $ b = -6 $, so:

→ $ x = \frac{-(-6)}{2(1)} = \frac{6}{2} = 3 $

Now plug into the equation to find y:

→ $ y = (3)^2 - 6(3) + 4 = 9 - 18 + 4 = -5 $

✔ Vertex: (3, -5)

Step 2: Axis of symmetry

→ Vertical line through vertex: $ x = 3 $

Step 3: Y-intercept

Set $ x = 0 $:

→ $ y = 0 - 0 + 4 = 4 $

✔ Y-intercept: (0, 4)

Step 4: X-intercepts (optional, if needed)

Solve $ x^2 - 6x + 4 = 0 $

Use quadratic formula:

→ $ x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(4)}}{2(1)} = \frac{6 \pm \sqrt{36 - 16}}{2} = \frac{6 \pm \sqrt{20}}{2} = \frac{6 \pm 2\sqrt{5}}{2} = 3 \pm \sqrt{5} $

→ Approximate: $ \sqrt{5} \approx 2.24 $ → x ≈ 0.76 and 5.24

✔ X-intercepts: approx (0.76, 0) and (5.24, 0)

Step 5: Plot points and sketch

- Vertex at (3, -5)
- Y-intercept at (0, 4)
- Use symmetry: since axis is x=3, point (0,4) reflects to (6,4)
- Add another point, say x=1: y = 1 -6 +4 = -1 → (1, -1); symmetric point x=5: y=25-30+4=-1 → (5,-1)

✔ Plot these points and draw a smooth U-shaped curve.

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## 🔹 Problem 2: $ y = x^2 + 2x - 5 $

Step 1: Vertex

$ x = \frac{-b}{2a} = \frac{-2}{2(1)} = -1 $

→ $ y = (-1)^2 + 2(-1) - 5 = 1 - 2 - 5 = -6 $

✔ Vertex: (-1, -6)

Step 2: Axis of symmetry → $ x = -1 $

Step 3: Y-intercept → x=0: y = 0 + 0 -5 = -5 → (0, -5)

Step 4: X-intercepts

Solve $ x^2 + 2x - 5 = 0 $

→ $ x = \frac{-2 \pm \sqrt{4 + 20}}{2} = \frac{-2 \pm \sqrt{24}}{2} = \frac{-2 \pm 2\sqrt{6}}{2} = -1 \pm \sqrt{6} $

→ $ \sqrt{6} \approx 2.45 $ → x ≈ 1.45 and -3.45

✔ X-intercepts: approx (1.45, 0) and (-3.45, 0)

Step 5: Additional points

Use symmetry: y-intercept (0, -5) reflects over x=-1 → (-2, -5)

Try x=1: y=1+2-5=-2 → (1,-2); symmetric point x=-3: y=9-6-5=-2 → (-3,-2)

✔ Plot vertex (-1,-6), y-int (0,-5), x-ints (~1.45,0) and (~-3.45,0), and other points. Draw parabola.

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## 🔹 Problem 3: $ y = x^2 - 3 $

This is a simple vertical shift of $ y = x^2 $

Step 1: Vertex

Since no x-term, vertex is at (0, -3)

✔ Vertex: (0, -3)

Step 2: Axis of symmetry → x=0 (y-axis)

Step 3: Y-intercept → (0, -3) — same as vertex

Step 4: X-intercepts

Solve $ x^2 - 3 = 0 $ → $ x^2 = 3 $ → $ x = \pm \sqrt{3} \approx \pm 1.73 $

✔ X-intercepts: approx (1.73, 0) and (-1.73, 0)

Step 5: Additional points

x=1 → y=1-3=-2 → (1,-2); symmetric point (-1,-2)
x=2 → y=4-3=1 → (2,1); symmetric point (-2,1)

✔ Plot these points and draw symmetric U-shape opening up from vertex (0,-3).

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## 🔹 Problem 4: $ y = x^2 + 6x + 1 $

Step 1: Vertex

$ x = \frac{-6}{2(1)} = -3 $

→ $ y = (-3)^2 + 6(-3) + 1 = 9 - 18 + 1 = -8 $

✔ Vertex: (-3, -8)

Step 2: Axis of symmetry → x = -3

Step 3: Y-intercept → x=0: y = 0 + 0 + 1 = 1 → (0,1)

Step 4: X-intercepts

Solve $ x^2 + 6x + 1 = 0 $

→ $ x = \frac{-6 \pm \sqrt{36 - 4}}{2} = \frac{-6 \pm \sqrt{32}}{2} = \frac{-6 \pm 4\sqrt{2}}{2} = -3 \pm 2\sqrt{2} $

→ $ \sqrt{2} \approx 1.41 $ → x ≈ -3 ± 2.82 → x ≈ -0.18 and -5.82

✔ X-intercepts: approx (-0.18, 0) and (-5.82, 0)

Step 5: Additional points

Use symmetry: y-int (0,1) reflects over x=-3 → (-6,1)

Try x=-1: y=1-6+1=-4 → (-1,-4); symmetric point x=-5: y=25-30+1=-4 → (-5,-4)

✔ Plot vertex (-3,-8), y-int (0,1), x-ints (~-0.18,0) and (~-5.82,0), and symmetric points. Draw parabola.

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## ✔ Summary for Graphing Each:

| Problem | Vertex | Axis of Symmetry | Y-intercept | Key Points to Plot |
|---------|--------------|------------------|-------------|---------------------------------------------|
| 1 | (3, -5) | x=3 | (0,4) | (0,4), (6,4), (1,-1), (5,-1) |
| 2 | (-1, -6) | x=-1 | (0,-5) | (0,-5), (-2,-5), (1,-2), (-3,-2) |
| 3 | (0, -3) | x=0 | (0,-3) | (1,-2), (-1,-2), (2,1), (-2,1) |
| 4 | (-3, -8) | x=-3 | (0,1) | (0,1), (-6,1), (-1,-4), (-5,-4) |

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## 📌 How to Actually Graph on Paper:

1. Draw your coordinate plane (already provided).
2. Mark the vertex — this is the lowest point (since a=1>0).
3. Draw the axis of symmetry as a dashed vertical line.
4. Plot the y-intercept.
5. Use symmetry to find the mirror point across the axis.
6. Plot 1–2 more points on each side to make sure the curve is accurate.
7. Connect the points smoothly with a curved line — don’t use straight lines!

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✔ You now have all the information needed to accurately graph each of the four quadratic functions on the provided grids.

Let me know if you’d like printable graph paper or digital graphing help!