https://file.aiquickdraw.com/mathbot/file/1729540341308rzvl1dhh.jpg

You provided a list of quadratic equations and a task to sketch their graphs. Here's a breakdown of the equations from the image:

1. **a)** \( y = x^2 - 6x + 8 \)
2. **b)** \( y = x^2 - 8x + 12 \)
3. **c)** \( y = x^2 + 8x + 15 \)
4. **d)** \( y = x^2 - 6x - 16 \)
5. **e)** \( y = x^2 - 2x - 8 \)
6. **f)** \( y = x^2 - 4x - 21 \)
7. **g)** \( y = x^2 + 8x + 7 \)
8. **h)** \( y = x^2 - 10x + 24 \)
9. **i)** \( y = x^2 - 12x + 20 \)

### Approach to sketching the graphs:
Each equation is a quadratic of the form \( y = ax^2 + bx + c \), representing a parabola. To sketch these:

1. **Find the vertex** using the formula:
   \[
   x_{vertex} = \frac{-b}{2a}
   \]
   Then substitute this value into the equation to find \( y_{vertex} \).

2. **Identify the direction** of the parabola:
   - If \( a > 0 \), the parabola opens upward.
   - If \( a < 0 \), the parabola opens downward.

3. **Find the y-intercept** by setting \( x = 0 \), and solving for \( y \).

4. **Find the x-intercepts** (roots) using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   If the discriminant \( b^2 - 4ac > 0 \), there are two real roots. If it equals 0, there is one real root. If it is negative, no real roots.

I'll start with the first few equations for you and provide details:

### a) \( y = x^2 - 6x + 8 \)
- \( a = 1 \), \( b = -6 \), \( c = 8 \)
- Vertex: \( x = \frac{-(-6)}{2(1)} = 3 \), \( y = 3^2 - 6(3) + 8 = -1 \)
- Direction: Opens upward since \( a = 1 > 0 \)
- y-intercept: \( y = 8 \) (when \( x = 0 \))
- x-intercepts: Using quadratic formula \( x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(8)}}{2(1)} = \frac{6 \pm \sqrt{36 - 32}}{2} = \frac{6 \pm 2}{2} \)
   - \( x = 4 \), \( x = 2 \)

### b) \( y = x^2 - 8x + 12 \)
- \( a = 1 \), \( b = -8 \), \( c = 12 \)
- Vertex: \( x = \frac{-(-8)}{2(1)} = 4 \), \( y = 4^2 - 8(4) + 12 = -4 \)
- Direction: Opens upward
- y-intercept: \( y = 12 \) (when \( x = 0 \))
- x-intercepts: \( x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(12)}}{2(1)} = \frac{8 \pm \sqrt{64 - 48}}{2} = \frac{8 \pm \sqrt{16}}{2} = \frac{8 \pm 4}{2} \)
   - \( x = 6 \), \( x = 2 \)

Would you like the analysis of the remaining quadratics, or more details on sketching?

Here are 5 questions related to quadratic graphing:
1. How does the value of \( a \) affect the direction of a parabola?
2. What does the vertex represent in a quadratic graph?
3. Why are x-intercepts important in graphing quadratic functions?
4. How does the discriminant influence the shape of a parabola?
5. Can a parabola have no x-intercepts? What does that signify?

**Tip**: Always start by finding the vertex, as it helps position the parabola on the graph accurately.

Sketch the graphs of the following quadratics: a) y = x^2 - 6x + 8, b) y = x^2 - 8x + 12, c) y = x^2 + 8x + 15, d) y = x^2 - 6x - 16, e) y = x^2 - 2x - 8, f) y = x^2 - 4x - 21, g) y = x^2 + 8x + 7, h) y = x^2 - 10x + 24, i) y = x^2 - 12x + 20.

Learn how to sketch graphs of quadratic equations like y = x^2 - 6x + 8 with step-by-step guidance on finding the vertex, direction, and intercepts. Perfect for high school algebra students.

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