Quadratic Functions
Section 01 – What is a Quadratic Function ?
A quadratic function is type of a polynomial function with a degree 2, expressed in its standard form ax2+bx+c , where a, b and c are real numbers [ a,b &c ∈ ℝ] and a ≠ 0. The highest exponent of the variable x is 2 which makes the function “quadratic” because “quad” means 2 or square.
y=x2 – This is called the parent quadratic function. All other types of quadratic functions we see are just transformations applied to the parent function.I will talk about transformations further in this lesson. For instance – in y=x2 + 1 , the +1 shifts the whole parabola up by 1 unit.
Quadratic Functions can exist in 3 different forms, and each form gives us some sort of information about how can we graph it,providing us distinct hints of graphical features : 1. Standard Form/General Form – y=ax2 +bx+c , here a indicates the direction the parabola If, if a is negative (a<0) , the parabola opens downward and if a is positive (a>0) , the parabola opens upward :
The standard form also tells us the y-intercept (where the parabola crosses the y-axis) which is the constant term ‘c‘. The graph crosses the y-axis when x is 0 , so the y intercept is (0,c). From here we can also see the value of the coefficients a , b and c . For example , in this quadratic function – f(x)=3x2 + 2x- 1 , a=3 , b=2 and c=-1 (*Remember we also have to take the numbers with their signs*). It is really useful when we need to use the quadratic formula, we need to plug in these values in the formula. The formula is discussed later in the lesson videos.
We can also find the vertex of the parabola from this form (*refer to the next section and lesson video where I discuss and break down everything about the quadratic graph*). For now, the vertex is the turning point of the parabola or the point where the graph reaches its maximum or minimum value. Whether the graph has a maximum or minimum point is dependent on which direction the parabola opens, which is controlled by ‘a‘ ( the coefficient of x2 ) as you can see above in the picture. If the parabola opens upward , then the graph has a minimum value , and if the parabola opens downward, the parabola has a maximum value.
The maximum or minimum points can also be called the stationary points of the parabola. We can find the x coordinate of the vertex by using this formula – -b/2a , and y coordinate can be found by substituting back into the equation. This vertex formula can also be proved by completing the square on the standard form (*discussed in Video lesson-3*).
2.Factored Form-y=a(x-r1) (x-r2) : from the factored form of the quadratic function,we can also determine the direction the parabola opens using the value of a. We can find the factors using zero product property which are the roots or x-intercepts of a quadratic function. The values of r1 and r2 are called zeroes (solutions) of the quadratic function.
The solutions to a quadratic function are also called “roots”. Any quadratic equation can maximum have 2 roots because its maximum degree is 2. Usually, in polynomial system, the degree decides the total possible number of roots. Another way to say this is , the quadratic roots are the x values where the graph crosses the x axis. At this point, the y value is 0.Basically, the solution to a quadratic equation can be called roots,zeroes,solutions or x-intercepts.
3.Vertex Form-y=a(x-h)2 +k : As you may have already guessed from the name, the vertex form helps us to easily identify the vertex. Remember, the vertex is turning point of a parabola. Depending on the direction the parabola opens the vertex can be the maximum or minimum point. The vertex is written as (h,k), it is where the parabola intersects its axis of symmetry.
If we know the vertex of the parabola, then we also have the axis of symmetry (AOS). Axis of symmetry is an imaginary line that divides the parabola into two equal halves, each halve is a mirror of each other,meaning if you folded the graph along the axis of symmetry, the 2 sides will overlap perfectly. It will always pass through the vertex of the parabola.(*refer to the Video lesson Part -2 to learn why a parabola is symmetric along AOS*)
We can also identify transformations of the parabola from this form. For example- the value of a determines the width of the parabola and the values h and k shows how the parent function y=x2 has been shifted. (*check out lesson video-2 for detailed explanation on these*)
