Vertex Form: y=(x-2)^2-5 Standard Form: y=x^2-4x-1 Advantages: See solution.
Practice makes perfect
We are told that the graph of y=x^2 is translated 2 units right and 5 units down. Let's begin by performing the translation. Note that, in the given function, we have that a=1.
A translation only affects the location of a graph. It does not affect the orientation nor the width of the parabola. Therefore, the value of a does not change and it will still be a = 1. With this, we can partially write the rule for the translated function in vertex form.
y= a(x- h)^2+k ⇔ y= 1(x- h)^2+k
In this form, ( h,k) is the vertex of the parabola. In our graph we can see that the vertex of the translated graph is ( 2,- 5). With this information, we can write the full equation in vertex form.
y= 1(x- 2)^2+(- 5) ⇔ y=(x-2)^2-5
An advantage of having the vertex form is being able to immediately identify the vertex of the parabola. Next, we will write the function in standard form. To do so, we will simplify the right-hand side of the above equation.
Now, we have a quadratic equation written in standard form.
y=ax^2+ bx+c
In this form, the constant c gives us the y-intercept of the parabola. Therefore, its advantage is being able to immediately determine the y-intercept.
