Contents
Contents
Quadratic equation of parabola: Vertex to Standard form
To know how to convert from Standard form to Vertex form, go to this link: How to convert standard form to vertex form
The equation of the parabola in Vertex form:
y = a(x-h)^{2} + k
Where, a, h, k are constants and real numbers, and a ≠ 0.
x and y are variables, where (x,y) represents a point on the parabola.
The equation of the parabola in Standard form:
y = ax^{2 }+ bx + c
Where, a, b, c are constants and real numbers, and a ≠ 0.
x and y are variables, where (x,y) represents a point on the parabola.
From vertex to standard form conversion:
Our sample parabola to solve:
y = 5 (x^{ }+ 1)^{2} + 5
(You can write the equations on the Graph plotter to see how the parabola looks like).
Unlike the Standard to Vertex conversion, we can convert from Vertex to Standard just in 2 steps (In standard conversion we followed a 6 step process).
Step 1: Simplify the binomial by multiplying by itself.
(This step is just the reverse of Step 5 of the Standard to Vertex conversion method).
y = 5 (x^{ }+ 1)^{2} + 5
⇒ y = 5 {(x+1) (x+1)} + 5
⇒ y = 5 (x^{2} + 1x + 1x +1) + 5
⇒ y = 5 (x^{2} + 2x +1) + 5
Step 2: Simplify the other numbers. The resultant equation is the standard form.
y = 5 (x^{2} + 2x +1) + 5
⇒ y = 5x^{2} + 10x + 5 + 5
⇒ y = 5x^{2} + 10x + 10
This is the standard form.
To check how to convert from standard form to vertex form for the same equation, go to the link, and check the sample parabola solved in the process: How to convert standard form to vertex form
Example (1):
y = (x+2)^{2} -6
The vertex form: y = (x+2)^{2} -6
⇒^{ }y = (x+2)(x+2)-6
⇒^{ }y = x^{2} + 2x + 2x + 4 – 6
⇒^{ }y=x^{2 }+ 4x – 2. This is the standard form.
To check how to convert from standard form to vertex form for the same equation, go to the link, and check Example (1): How to convert standard form to vertex form
Example (2):
y = – (x^{ }-3)^{2} + 14
The vertex form: y = – (x^{ }-3)^{2} + 14
⇒^{ }y = – {(x^{ }-3) (x-3)} + 14
⇒^{ }y = – {(x^{ }-3) (x-3)} + 14
⇒^{ }y = – {(x^{2}– 3x – 3x + 9)} + 14
⇒^{ }y = – {(x^{2 }– 6x + 9)} + 14
⇒^{ }y = – x^{2 }+ 6x – 9 + 14
⇒^{ }y = – x^{2 }+ 6x + 5. This is the standard form.
To check how to convert from standard form to vertex form for the same equation, go to the link, and check Example (2): How to convert standard form to vertex form
Why convert from Vertex to Standard form?
- It is easier to understand the curve in Standard form (where it is easier to sketch the curve in Vertex form).
- It is easier to compute derivatives in the Standard form.
Understanding the curve using Standard form equation:
In standard form, the equation is simply factored, therefore, it is easy to find the x-intercepts and other information of the graph easily.
Let’s examine how different values of a, b, c impacts the parabola using the standard form.
Again repeating from above, the equation of the parabola in Standard form:
y = ax^{2 }+ bx + c
Where, a, b, c are constants and real numbers, and a ≠ 0.
x and y are variables, where (x,y) represents a point on the parabola.
“x” represents the unknown variable which can be found using the equation (x = -b/2a); from there the x value can be substituted into the equation to find “y”. Which we already illustrated in our other topic: How to convert standard form to vertex form
Our solved equation from the process:
y = 5x^{2} + 10x + 10 (Standard form).
Therefore, a = 5, b = 10, c = 10
- First, start with analyzing the value of “a”.
- “a” intercept tells you the vertical stretch of the graph, and the direction the parabola is facing.
- If |a|>1 the graph is vertically stretched.
- If 0<|a|<1, the graph is vertically shrunk.
- If a>0, the parabola faces upwards.
- If a<0, the parabola faces downwards.
- “a” intercept tells you the vertical stretch of the graph, and the direction the parabola is facing.
(The images here done by using Graph plotter)
Original Equation y = 5x^{2} + 10x + 10a = 5 | Changing a= ⅕, So the equation is now,y = x^{2} / 5 + 10x + 10Check the change in the graph: |
Changing a=-5, So the equation is now,y = -5x^{2} + 10x + 10Check the change in the graph: | |
- The “b” value translates the parabola horizontally across the x-axis.
- If the “b” value is positive, the parabola moves to the left.
- If the “b” value is negative, it moves to the right.
Original Equation y = 5x^{2} + 10x + 10b = 10 | Changing b = -10, So the equation is now,y = 5x^{2} – 10x + 10Check the change in the graph: |
- The “c” value represents the y-intercept, where the parabola crosses the y-axis.
Original Equation y = 5x^{2} + 10x + 10c = 10 | Changing c = -10, So the equation is now,y = 5x^{2} + 10x – 10Check the change in the graph: |