To rewrite $y = (x + 3)(x - 5)$ in vertex form, we’ll first expand and then complete the square.

Step 1: Expand the Expression

$y = (x + 3)(x - 5)$ Use the distributive property (FOIL method): $y = x^2 - 5x + 3x - 15$ $y = x^2 - 2x - 15$

Step 2: Complete the Square

Now that we have $y = x^2 - 2x - 15$, we’ll complete the square to convert it into vertex form.

1. Start with the expression involving $x$: $x^2 - 2x$.
2. Take half of the coefficient of $x$ (which is $-2$), square it, and add and subtract this value inside the equation: $y = (x^2 - 2x + 1) - 1 - 15$ $y = (x - 1)^2 - 16$

Step 3: Write in Vertex Form

The equation in vertex form is: $y = (x - 1)^2 - 16$

Vertex and Points

The vertex of this parabola is $(1, -16)$. We need at least two more points to graph it.

Finding Additional Points

1. Choose $x = 0$: $y = (0 - 1)^2 - 16 = 1 - 16 = -15$ So, $(0, -15)$ is one point.

2. Choose $x = 2$: $y = (2 - 1)^2 - 16 = 1 - 16 = -15$ So, $(2, -15)$ is another point.

Summary

• Vertex: $(1, -16)$
• Additional Points: $(0, -15)$ and $(2, -15)$

You can now plot these three points: $(1, -16)$, $(0, -15)$, and $(2, -15)$. The parabola will open upward because the coefficient of the squared term $(x - 1)^2$ is positive.

Would you like more details or have any questions?

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Related Questions:

1. How do you find the vertex of a parabola in standard form?
2. What are the steps to expand and factor quadratic expressions?
3. How do you determine if a parabola opens upwards or downwards?
4. How does completing the square help in converting to vertex form?
5. How would you find the axis of symmetry for this parabola?

Tip: Always check the vertex as it often gives the minimum or maximum value for a quadratic function, depending on the direction it opens.