How do you convert vertex form to slope-intercept form?

To convert a quadratic equation from vertex form to slope-intercept form, you need to start with the vertex form of the equation, which is given as:

y = a(x – h)² + k

In this form, (h, k) represents the vertex of the parabola, and ‘a’ indicates the direction and width of the parabola.

Here’s a step-by-step guide on how to make the conversion:

1. Expand the equation: Start by expanding the squared term. You’ll distribute the ‘a’ across the expanded expression.
2. Combine like terms: After expanding, combine all like terms to simplify the equation.
3. Rearrange into slope-intercept form: The slope-intercept form of a linear equation is y = mx + b, where ‘m’ is the slope and ‘b’ is the y-intercept. You will arrange your expanded equation to match this format.

For example, if you have:

y = 2(x – 3)² + 4

1. Expand it: y = 2((x – 3)(x – 3)) + 4 = 2(x² – 6x + 9) + 4

2. Distribute ‘a’: y = 2x² – 12x + 18 + 4

3. Combine the constant terms: y = 2x² – 12x + 22

You see that while the final expression is a quadratic, in order to have a linear format, it’s not possible as quadratics don’t fit into slope-intercept form unless simplified to a specific x-value. But for the specific linear segments in a parabola, finding the slope of the tangent line at any given point would comply to that form.

So, keep in mind your end goal when converting the equations and always look for the specific values you’re interested in!