Math Revision 15.1 - Quadratic Graphs Part One

In these revision notes for Quadratic Graphs Part One, we cover the following key points:

• What are quadratic equations? Why are they called so?
• What are the two types of quadratic equations?
• How do we solve quadratic equations?
• What shape does a quadratic graph have?
• How do we plot the graph of a quadratic equation?
• How do we make the distinction between various types of quadratic graphs?
• What are the special points we can use to plot a quadratic graph?
• What does the sign of the coefficient a indicate for quadratic graphs?
• How do we find the equation of a parabola from its graph?

Quadratic Graphs Part One Revision Notes

A quadratic equation with one variable - as the name suggests - is a type of equation that contains a single variable raised to a maximum power of 2. The variables in the first power are also present in the same equation in many cases. We usually express the variable by the letter x but other letters can also be used. The general form of such equations is

where a and b are coefficients, c is a constant and x is the variable.

A quadratic equation with two variables is a kind of equation where the independent variable (usually x) is at the second power at maximum, while the dependent variable (usually y) is always written to the first power. The general form of such equations is

Second-order equations with two variables are also known as quadratic functions. They are called "quadratic" because "quadratum" is the Latin word for "square" and since the independent variable is raised in the square, we call such equations quadratic.

The shape of quadratic graphs is a parabola. This is because, in geometry, the equation of a parabola has the basic form y = x². All the other elements added in the formula, such as any coefficient a preceding the variable x, any bx term or any constant c, simply make the parabola shift, stretch, compress or flip vertically with respect to the basic form y = x².

The most carefree way to plot a quadratic graph is to give some values to the independent variable x (the more points the better) and find the corresponding y-values. All points found are inserted in the proper place in the coordinates system and after this, they are connected smoothly. If the points are chosen randomly, the number of points must be at least 10 to 15. However, there are some special points that, if found, reduce the number of points needed to form a quadratic graph up to five. These five special points are:

1. Two x-intercepts (if discriminant is positive), with coordinates A(x₁, 0) and B(x², 0), where x₁ = -b - √∆/2a and
   x₂ = -b + √∆/2a
   Therefore, the coordinates of points A and B are
   A(-b - √∆/2a,0)
   and
   B(-b + √∆/2a,0)
2. The vertex point V. It has the coordinates x_V = -b/2a and
   y_V = -∆/4a
   Thus, we have for the vertex V:
   V(-b/2a,-∆/4a)
3. Another point of the parabola that is easy to find is the y-intercept. Since the x-coordinate of any y-intercept is always zero, we can solve the original quadratic equation for x = 0. In this way, we obtain y = c. Therefore the coordinates of the y-intercept (say point C) are C(0, c).
4. You can also find the symmetrical point to the y-intercept (say point D) to obtain a kind of symmetry in the points and therefore to make the graph more accurate. For this, we can again use the half-segment formula, where x_V is the half-segment point, x = 0 is one of the endpoints and the other endpoint has to be found. Using the midpoint formula for the direction x,
   x_V = x_C + x_D/2

we first find the x-coordinate of point D (x_D); then, we find the corresponding y-value by substituting x_D in the original equation, i.e.

If discriminant Δ is zero or negative, we can't make use of points A and B as when the denominator is zero, the graph touches the horizontal axis at a single point, which corresponds to the vertex V, while when the discriminant is negative, the graph does not touch the horizontal axis at all, so there are no x-intercepts. Therefore, points A and B are replaced by two other points that have the same horizontal distance from the vertex in either side of it.

The sign of the coefficient 'a' is very important in understanding the direction of a parabola formed by a quadratic equation. Thus, if this coefficient is positive, the parabola has the arms up, as occurred in all examples we have discussed so far. In such cases, the parabola has a minimum, which corresponds to the vertex V. On the other hand, when the coefficient 'a' is negative, the arms of the parabola are directed downwards and the parabola has a maximum at the vertex point V. The solution is the same as usual.

When the graph of a parabola is given and we want to find its equation, we (again) use the five special points to extract any useful information from them regarding the coefficients and the constant of the parabola.