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To sketch the graphs of the given quadratic equations, we need to analyze each one and determine important features such as the vertex, axis of symmetry, and intercepts. Each quadratic function is in the standard form \( y = ax^2 + bx + c \), and we will apply these steps for each:

1. **Find the vertex** using the formula for the x-coordinate of the vertex, \( x = \frac{-b}{2a} \), and then substitute this value back into the equation to find the y-coordinate.
2. **Identify the axis of symmetry**, which is the vertical line passing through the vertex, given by \( x = \frac{-b}{2a} \).
3. **Determine the y-intercept**, which is the value of \( y \) when \( x = 0 \), equal to \( c \).
4. **Find the x-intercepts (if any)** by solving \( ax^2 + bx + c = 0 \) using the quadratic formula if necessary.

Let’s start with one of the equations:

### (a) \( y = x^2 - 6x + 8 \)

1. **Vertex**:  
   The x-coordinate is \( x = \frac{-(-6)}{2(1)} = \frac{6}{2} = 3 \).  
   Substitute \( x = 3 \) into the equation to find \( y \):  
   \( y = (3)^2 - 6(3) + 8 = 9 - 18 + 8 = -1 \).  
   So, the vertex is \( (3, -1) \).

2. **Axis of symmetry**:  
   The axis of symmetry is \( x = 3 \).

3. **Y-intercept**:  
   When \( x = 0 \), \( y = 8 \).  
   So, the y-intercept is \( (0, 8) \).

4. **X-intercepts**:  
   Solve \( x^2 - 6x + 8 = 0 \).  
   Factor: \( (x - 2)(x - 4) = 0 \), so the x-intercepts are \( x = 2 \) and \( x = 4 \).

You can repeat these steps for each equation.

Would you like to go through all of them or need help with a specific one?

Here are five related questions to expand on this:

1. How do you derive the quadratic formula, and when is it necessary to use it?
2. What is the significance of the axis of symmetry in a parabola's graph?
3. How can the discriminant of a quadratic equation tell you about the number of real roots?
4. What happens to the graph of a parabola when you change the values of \( a \), \( b \), and \( c \)?
5. How would you graph a quadratic if it cannot be factored easily?

**Tip**: Always check whether you can factor a quadratic equation before applying the quadratic formula, as factoring is usually quicker.

Sketch the graphs of the following quadratics: y = x^2 - 6x + 8, y = x^2 - 8x + 12, y = x^2 + 8x + 15, y = x^2 - 6x - 16, y = x^2 - 2x - 8, y = x^2 - 4x - 21, y = x^2 + 8x + 7, y = x^2 - 10x + 24, y = x^2 - 12x + 20.

Learn how to graph quadratic equations like y = x^2 - 6x + 8 and others by finding key features such as vertices, intercepts, and axes of symmetry. This lesson covers the quadratic formula and factoring for Grades 9-11.

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