This week in Precalc 11, we learned many new things and started preparing for our unit test and midterm. One of the most interesting things we learned this week was equivalent forms of the quadratic function. From what we learned, we know 3 different types of forms …

Standard form : $y = a(x - p)^2 + q$

General form : $y = ax^2 + bx + c$

Factored form : $y = a(x - x_1)(x - x_1)$

Out of all these forms, I like standard form (vertex form) the best because to me its very simple to figure out the vertex and other things with standard form making it easier to graph the graph.

For example if we have : $y = (x + 1)(x - 11)$ (factored form)

We can distribute and will give us general form : $x^2 - 4x - 77$

When we figure out the equation in general form, we can now complete the square (which we learned in the previous unit) and figure the equation out in standard form which gives us : $y = (x - 2)^2 - 81$

Looking at the equation : $y = (x - 2)^2 - 81$ I know that it will be congruent to the parent function and the vertex is $(2, -81)$

When I’m practicing, I like using desmos to check if all the equations line up, when I’ve seen that they line up, I know I have done the algebra properly. Learning about equivalent equations are really important in case a question asks us to give information about a graph but it is in general form (which won’t give us much information about the graph). A question could also ask about the intercepts which we will use factored form for.