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11.26.2018: One Lesson of Math - Quadratic Function in Vertex Form, Part 2

  • Writer: Steven C. Mills
    Steven C. Mills
  • Nov 27, 2018
  • 2 min read

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:

  1. Compare the equation with its formula, y = a(x - p)² + q, so that we know which variable each number represents

  2. 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

  3. p and q are the y and x coordinates of the vertex

  4. p is the axis of symmetry

  5. Solve for y with x=0 to find the y-intercept

  6. Use the quadratic formula to find the x-intercepts

  7. The domain is always x ∈ ℝ

  8. 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:

  1. Set up the vertex formula, y = a(x - p)² + q

  2. 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.

  3. 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

  4. 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.

  5. 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.

  6. 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.

  7. 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

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