Sketching Quadratic Graphs - Free Printable

You’re asked to sketch the graphs of four quadratic functions, and your sketch should identify:

- The roots (x-intercepts)
- The y-intercept
- The turning point (vertex)
- And be drawn freehand (so no need for perfect precision, but key features must be clear)

Let’s go through each one step-by-step.

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## 🔹 a) \( y = x^2 + 6x + 8 \)

Step 1: Factor to find roots

\( y = x^2 + 6x + 8 = (x+2)(x+4) \)

→ Roots: \( x = -2 \), \( x = -4 \)

Step 2: Find y-intercept

Set \( x = 0 \):
\( y = 0 + 0 + 8 = 8 \) → y-intercept: \( (0, 8) \)

Step 3: Find turning point (vertex)

For \( y = ax^2 + bx + c \), vertex x-coordinate is \( x = -\frac{b}{2a} \)

Here, \( a=1, b=6 \) → \( x = -\frac{6}{2(1)} = -3 \)

Plug into equation:
\( y = (-3)^2 + 6(-3) + 8 = 9 - 18 + 8 = -1 \)

→ Vertex: \( (-3, -1) \)

Step 4: Sketch

- Opens upward (since \( a = 1 > 0 \))
- Roots at \( x = -4 \) and \( x = -2 \)
- y-intercept at \( (0, 8) \)
- Vertex at \( (-3, -1) \)

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## 🔹 b) \( y = x^2 + 4x - 12 \)

Step 1: Factor to find roots

\( y = x^2 + 4x - 12 = (x+6)(x-2) \)

→ Roots: \( x = -6 \), \( x = 2 \)

Step 2: y-intercept

Set \( x = 0 \):
\( y = 0 + 0 - 12 = -12 \) → y-intercept: \( (0, -12) \)

Step 3: Vertex

\( x = -\frac{b}{2a} = -\frac{4}{2(1)} = -2 \)

\( y = (-2)^2 + 4(-2) - 12 = 4 - 8 - 12 = -16 \)

→ Vertex: \( (-2, -16) \)

Step 4: Sketch

- Opens upward
- Roots at \( x = -6 \), \( x = 2 \)
- y-intercept at \( (0, -12) \)
- Vertex at \( (-2, -16) \)

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## 🔹 c) \( y = x^2 + 10x - 4 \)

This doesn’t factor nicely → use quadratic formula for roots.

Step 1: Roots via quadratic formula

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a=1, b=10, c=-4 \)

Discriminant: \( D = 10^2 - 4(1)(-4) = 100 + 16 = 116 \)

So,

\( x = \frac{-10 \pm \sqrt{116}}{2} = \frac{-10 \pm 2\sqrt{29}}{2} = -5 \pm \sqrt{29} \)

√29 ≈ 5.385 → so roots approximately:

→ \( x ≈ -5 + 5.385 = 0.385 \)
→ \( x ≈ -5 - 5.385 = -10.385 \)

Step 2: y-intercept

Set \( x = 0 \):
\( y = 0 + 0 - 4 = -4 \) → y-intercept: \( (0, -4) \)

Step 3: Vertex

\( x = -\frac{10}{2(1)} = -5 \)

\( y = (-5)^2 + 10(-5) - 4 = 25 - 50 - 4 = -29 \)

→ Vertex: \( (-5, -29) \)

Step 4: Sketch

- Opens upward
- Roots approx. at \( x ≈ -10.4 \) and \( x ≈ 0.4 \)
- y-intercept at \( (0, -4) \)
- Vertex at \( (-5, -29) \) — very low!

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## 🔹 d) \( y = 4 - x - 2x^2 \)

Rewrite in standard form:
→ \( y = -2x^2 - x + 4 \)

Note: Leading coefficient is negative → parabola opens downward

Step 1: Roots via quadratic formula

\( a = -2, b = -1, c = 4 \)

Discriminant: \( D = (-1)^2 - 4(-2)(4) = 1 + 32 = 33 \)

Roots:

\( x = \frac{-(-1) \pm \sqrt{33}}{2(-2)} = \frac{1 \pm \sqrt{33}}{-4} \)

√33 ≈ 5.744

So:

→ \( x ≈ \frac{1 + 5.744}{-4} = \frac{6.744}{-4} ≈ -1.686 \)
→ \( x ≈ \frac{1 - 5.744}{-4} = \frac{-4.744}{-4} ≈ 1.186 \)

Step 2: y-intercept

Set \( x = 0 \):
\( y = 4 - 0 - 0 = 4 \) → y-intercept: \( (0, 4) \)

Step 3: Vertex

\( x = -\frac{b}{2a} = -\frac{-1}{2(-2)} = \frac{1}{-4} = -0.25 \)

Plug into equation:

\( y = -2(-0.25)^2 - (-0.25) + 4 \)
= \( -2(0.0625) + 0.25 + 4 \)
= \( -0.125 + 0.25 + 4 = 4.125 \)

→ Vertex: \( (-0.25, 4.125) \)

Step 4: Sketch

- Opens downward
- Roots approx. at \( x ≈ -1.69 \) and \( x ≈ 1.19 \)
- y-intercept at \( (0, 4) \)
- Vertex at \( (-0.25, 4.125) \) — slightly above y-intercept

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## ✔ Final Sketching Tips

When sketching freehand:

- Draw the U-shape (or inverted U) with smooth curve.
- Mark roots on x-axis.
- Mark y-intercept on y-axis.
- Mark vertex — highest or lowest point.
- Label axes and key points if possible.

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## 📌 Summary Table

| Function | Roots (approx.) | y-intercept | Vertex (approx.) | Direction |
|-------------------|--------------------------|-------------|------------------------|---------|
| a) \( x^2+6x+8 \) | \( x = -4, -2 \) | (0, 8) | (-3, -1) | Up |
| b) \( x^2+4x-12 \)| \( x = -6, 2 \) | (0, -12) | (-2, -16) | Up |
| c) \( x^2+10x-4 \)| \( x ≈ -10.4, 0.4 \) | (0, -4) | (-5, -29) | Up |
| d) \( -2x^2-x+4 \)| \( x ≈ -1.69, 1.19 \) | (0, 4) | (-0.25, 4.125) | Down |

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✔ You now have all the information needed to sketch each graph accurately! Just plot the key points and draw a smooth parabola through them. Let me know if you’d like hand-drawn sketches or graph images — I can describe how to draw them step by step too.