Today's soundtrack is Christmas With the Mills Family, a Spotify playlist that includes my favourite Christmas songs.
I've already learned how to use the quadratic formula to work with the general form of the quadratic equation; after that, I learned how to determine the characteristics of a quadratic function in the vertex form. Today, I'll be learning how to determine characteristics of a quadratic function from its general form.
The general (or "expanded") form of a quadratic function is y = ax² + bx + c.
We can learn the following characteristics of the graph of a quadratic function in general form:
The y-intercept
the constant at position c is the y-intercept
The x-intercept(s), if there are any, by factoring or using the quadratic formula
The axis of symmetry, if the function has x-intercepts, or even a single x-intercept
Logically, the axis of symmetry is between the x-intercepts: (x1 + x2)/2 = axis
If there is a single x-intercept, the axis of symmetry is on the x-intercept
The coordinates of the vertex, if the axis of symmetry is known
Since the x-coordinate of the vertex is the axis of symmetry, if we already know the axis of symmetry (see point the third), we are only missing the y-coordinate.
To find the y-coordinate, we use substitution: by substituting the variable x in the original equation with the constant which is the axis of symmetry, we can solve for y and find the y-coordinate of the vertex