Today's soundtrack is Living Sacrifice: Conceived in Fire, one of my favourite Christian metal albums.
This afternoon, I'm continuing on to the second portion of the lesson I just did about the vertex form of the quadratic function.
To graph a quadratic function from its vertex form, we do the following:
Compare the equation with its formula, y = a(x - p)² + q, so that we know which variable each number represents
If a is negative, the vertex will be the function's maximum point; if a is positive, the vertex will be the function's minimum point
p and q are the y and x coordinates of the vertex
p is the axis of symmetry
Solve for y with x=0 to find the y-intercept
Use the quadratic formula to find the x-intercepts
The domain is always x ∈ ℝ
The range depends on whether a is positive or negative. If negative, y ≤ [vertex's y value], y ∈ ℝ. If positive, y ≥ [vertex's y value], y ∈ ℝ.
To determine the equation of a quadratic function based on its axis of symmetry and two points that it passes through, we can extrapolate its equation by doing the following:
Set up the vertex formula, y = a(x - p)² + q
Since the axis of symmetry's equation is x = p, we insert the axis of symmetry in for p, keeping in mind that if the axis is negative, we will insert it as a positive, since we are putting it in at -p.
Next, we find the values of a and q through a multi-step process: first, rewrite the equation, y = a(x - p)² + q, in two columns - including the new value of p as discussed in step 2
In the first column, substitute our first coordinate's y-value into y in the equation, and substitute its x-value into x. Do the same in the second column with the second coordinate's values.
We should now have two equations, both of which still do not have values for a or q. We can now solve these by using substitution. First, isolate a in the first equation, then isolate q in the second equation.
After isolating the variables, we substitute the value of q in the second equation into the variable q in the first equation, and we substitute the value of a in the first equation into the variable a in the second equation. We now have the true values of a and q.
Substitute the true values of a and q into our equation y = a(x - p)² + q, retaining the value of p as found in step 2. We now have our full equation with only the variables y and x left as variables!
A quick vertex formula cheat sheet to find the vertex coordinates:
At the start of the equation:
- [number] means that the vertex is the maximum point
[number] means that the vertex is the minimum point
Inside the brackets:
x - [number] means move [number] of spaces to the right
x + [number] means move [number] of spaces to the left
At the end of the equation:
- [number] means move down [number] of spaces
+ [number] means move up [number] of spaces