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11.22.2018: One Lesson of Math - Properties of a Quadratic Function, Part 2


Petra: Jekyll & Hyde

Today's soundtrack is Petra: Jekyll and Hyde, which I still thought of as being a fairly recent album until I looked up the artwork for today's post and realized that it was released in 2003. It's fifteen years old. I bought it shortly after it was released. Time flies when you're having fun, I guess.


Unrelated side note: My website has hit a new milestone today - 200 unique visitors!


This afternoon, I'm working on the second part of the quadratic function lesson. In this unit, I will need to identify the characteristics of a quadratic function from its graph. I'm supposed to determine the following:

"i) the x- and y-intercepts

ii) the coordinates of the vertex

iii) the equation of the axis of symmetry

iv) the domain of the function

v) the range of the function" (Pearson, Pre-Calculus 11, p. 252)


When answering questions about coordinates, we give answers in the following format: (x value, y value). For example, if a quadratic function's vertex is 12 on the y-axis and -4.5 on the x-axis, we would say that the vertex is at (-4.5, 12).


The x- and y-intercepts are the places at which the parabola's curve crosses either the x-axis or the y-axis (any place where we see that one of the two coordinates is 0).


The coordinates of the vertex are the place where the parabola is at either its highest or lowest y-value, as the case may be.


The equation of the axis of symmetry is the x-value on which the vertical line of symmetry lies.


The domain is "all possible x-values" (p. 253), and since quadratic functions are infinite, is always x ∈ ℝ.


The range is "all possible y-values" (p. 253), and will be finite in one direction and infinite in the other. For example, in the case of a negative ax quadratic function whose vertex is at (-1, 3), its domain would be y ≥ -1, y ∈ ℝ.


The next topic that I'll be using as I analyze quadratic functions is called first differences. After plugging x and y values of a graph into a table, we can determine whether a function is linear or quadratic by subtracting each y value from the one immediately below it. If the differences are constant (3, 3, 3, 3...), the function is linear; if we see progressive changes (-3, -1, 1, 3, 5, 7...), we know the function is quadratic.


If we are presented with a quadratic equation that uses fractions or decimals, my textbook advises that we "use technology to graph them" (p. 255).

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