Jargon for Quadratic Curve Sketching
Prerequisite Knowledge for Quadratic Curve Sketching
1. Lower Secondary Algebraic Manipulation
2. Quadratic Equation Solving
3. Coordinate Geometry
What are Quadratic Curves?
Quadratic curves are parabolas, meaning that they are plane curves which are mirror-symmetrical and are approximately U-shaped. Four examples plotted using graphing software are shown below:
Quadratic curves have the following characteristics:
Highest power of the independent variable is 2 (independent variable is usually denoted by x, while the dependent variable is usually denoted by y);
Perfectly symmetrical about a vertical line;
U-shaped;
Extend infinitely in two directions; and
Have one turning point each.
Every quadratic curve can be represented by a formula where y is the subject of the equation. Such formulas that represent quadratic curves come in two formats: the generic format and the completed square format. These are defined below:
You need to be familiar with both to be able to sketch all possible quadratic curves. Both formats are useful for different purposes and have their own unique limitations. The generic format is most commonly given and is useful for determining the quadratic curve’s shape (be it a “happy-face” or “unhappy-face”), easily calculating the discriminant [(b^2) – 4ac], and obtaining the graph’s y-intercept (the constant term, denoted by c). However, the generic format is not so useful for determining the coordinates of the curve’s turning point, which students are expected to calculate and label (unless you are familiar with calculus, in which case the generic format could be used this purpose).
This is where the completed square format comes in. The completed square format (y = a(x – h)^2 + k) clearly shows the values which h and k take on, which are the x and y coordinates of the turning point respectively (the explanation for this is given below in step 6). This format can also be useful for finding x-intercepts, as solving the equation when y = 0 is easier since the “square” is already “complete”. The completed square format is not useful for much else. Also, it can be difficult to begin sketching the graph with this format (unless you use concepts taught in ‘A’ level mathematics). Neither does this format state the y-intercept.
If you are provided the formula of the quadratic curve in the generic format, you can convert it into the completed square format by … …(you guessed it,) executing procedures for completing the square. Whereas if you are provided the same thing but in the completed square format, you can convert it into the generic format by carrying out algebraic expansion.
Steps for Sketching Quadratic Curves:
Determine the type of graph to be sketched (linear, quadratic, or cubic?).
Ascertain the type of quadratic graph to be sketched (does the shape take on a “happy face” or “unhappy face”? How many intercepts are there?).
Calculate x-intercept(s) (if any).
Determine the location of the y-axis relative to the graph.
Calculate y-intercept.
Ascertain coordinates of turning point.
Label the Graph’s Significant Characteristics
The breakdown of each step is provided in the sections below.
Step 1: Determine the Type of Graph to be Sketched
If you were to climb a ladder, you should first seek to determine whether the ladder is resting against the correct wall in the first place before committing to climbing it. Similarly, you should first seek to determine whether the graph to be sketched is quadratic before rushing to execute all the “Steps for Sketching Quadratic Curves”. Hence the purpose of step 1.
In the ‘O’ Level syllabus for Elementary Mathematics, you would come across at least three types graphs. They are set out below:
A graph is a quadratic graph if the highest power of x in its formula is 2. Proceed with the remaining steps ONLY IF the graph to be sketched is QUADRATIC!
Step 2: Ascertain the Type of Quadratic Curve
Quadratic graphs come in two general shapes: a “happy face” and an “unhappy face”. A quadratic graph takes on a “happy face” when its coefficient of x^2 is positive, and an “unhappy face” when it is negative.
Next, the number of real roots (x-intercepts) must be determined. This can be ascertained by calculating the discriminant [(b^2) – 4ac] and evaluating whether the discriminant is positive, negative, or equal to zero. A quadratic curve has two real roots if the discriminant is positive, one real root if the discriminant is equal to zero, and no real roots if the discriminant is negative.
From the above, it can be deduced that there are six possible types of quadratic curves. These are set out below:
Step 3: Calculate x-intercept(s) (if any)
x- intercepts are the points on the graph where the graph intersects the x-axis. The exact points are determined by equating y as zero. This is because the equation which defines the x-axis is y = 0. Another perspective is to realize that every single point on the x-axis has a y-coordinate of zero, hence equating y as zero in the equation will produce the x-coordinates of the x-intercepts.
In equating y as zero, you will have something like this:
When y = 0, a(x^2) + bx + c = 0
Solving this quadratic equation will give you the x-intercepts. If the discriminant is positive, solving the quadratic equation should give you two answers for x. If the discriminant is equal to zero, solving the quadratic equation should give you one answer for x.
However, if the discriminant is negative, do not bother solving the quadratic equation as it would be impossible to solve without using concepts of complex numbers. In such a scenario, the quadratic curve has no real roots and (as far as the 'O' Level syllabus is concerned) this step should be skipped.
Step 4: Determine Location of the y-axis Relative to the Curve
The location of the y-axis can be deduced from either the graph’s x-intercepts or the x-coordinate of its turning point.
x- coordinates on the right of the y-axis are positive while those on the left are negative. As such, if an x-intercept is positive, the y-axis is to the left of that intercept. If an x-intercept is negative, then the y-axis is on the right of that intercept. These are illustrated below:
(Note: assumptions were made that a > 0 and (b^2) – 4ac > 0. If assumption(s) is(are) false, change your sketch accordingly.)
If the graph has both positive and negative intercepts, then there are three possibilities. These are illustrated below:
(Note: assumptions were made that a > 0 and (b^2) – 4ac > 0. If assumption(s) is(are) false, change your sketch accordingly.)
If x-coordinate of turning point is positive, then the y-axis is to the left of the turning point. If it is negative, then the y-axis is to the right of the turning point. If it is zero, then the y-axis passes through the turning point.
Step 5: Calculate y-intercept
y-intercepts are points on the graph where the graph intersects the y-axis. Upright quadratic curves will each have only one y-intercept. This exact point is determined by equating x as zero. The reason for this is that x = 0 is the equation which defines the y-axis. Another perspective is to realize that every point on the y-axis has an x-coordinate of zero, hence equating x as zero in the equation would produce the y-coordinate of the y-intercept.
In equating x as zero, you will have something like this:
When x = 0, y = a[(0)^2] + b(0) + c
y = 0 + 0 + c
y = c
Considering the working above, it is easy to see why the constant term of the generic formula (the term “c” in y = a(x^2) + bx + c) is also called the y-intercept.
Step 6: Ascertain Coordinates of Turning Point
First, obtain the equation of the curve in the form of the completed square format. If you have only the generic format, use procedures for completing the square to convert it into the completed square format.
Once you have obtained the equation in the form of: y = a[(x – h)^2] + k, you can then state that the coordinates of the curve’s turning point are ( h , k ). For example, for the graph y = 2[(x + 3)^2] – 4, the coordinates of its minimum point are (-3, -4).
Step 7: Label the Graph’s Significant Characteristics
The significant characteristics of the graph which need to be labelled are listed below:
(Note: assumptions were made that a > 0 and b^2 – 4ac > 0. If assumption(s) is(are) false, change your sketch accordingly.)
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