Helpful Tools for Calculus, Chapter 1: Functions and Relations
Previous Section: 1.2 Linear Functions
Immediately following linear functions in terms of difficulty are Quadratic functions. These are the most simple of the curved functions and work as a solid playground for developing new ideas. This section will introduce quadratics and a variety of important base information about them.
Forms
A Quadratic Function is a function defined by a quadratic polynomial. This generally presented as y=ax2+bx+c with constants a,b, and c where a≠0. Here, c represents the y-intercept.
Another method of writing a quadratic is in factored form. This is not always possible, because a quadratic cannot always be factored. However, if we can rewrite the equation as y=a(x-h)(x-k) then it is easy to see the x-intercepts which occur at x=h and x=k. The last method of writing a quadratic formula stems from conic sections because a parabola is one of the possible shapes created when slicing a cone. When using the technique of Completing the Square we would write the quadratic in the following form:
Sometimes this is presented as: y=a(x-h)2+k. This form is helpful because it helps us to identify the center of the parabola which occurs at 
. Because of this, it is often called Vertex Form. The method of Completing the Square is used to get this and will be expanded on later in this section.
Example 1: Factor the following equation and identify its x and y intercepts: y=2x2+4x-30
Solution:
So y=2x2+4x-30 has x-intercepts at x=-5, 3.
Example 2: Where is the center (vertex) of the following parabola? Be sure to write it as a coordinate pair. y=3(x-4)2+5
Solution: This matches the vertex form of the equation, so the value can be easily identified from the equation itself: (4,5)
Example 3: Where is the center (vertex) of the following parabola? Be sure to write it as a coordinate pair. y=x2-3x+2
Solution: In this form, the easiest method is to label the a, b, and c values then use the formulas that define the vertex method:
Completing the Square
The method of Completing the Square is useful for taking a quadratic from the standard form, y=ax2+bx+c, to the vertex form, y=a(x-h)2+k. The way that this happens is by factoring a out of the terms with x, then identifying what constant would make the resulting quadratic a perfect square trinomial. That is to say, we identify how can me make it a perfect square even if something else is added to the end.
This is done by splitting the equation up as follows:
We then take the term on x and divide it by 2 before squaring it and adding it to both sides.
The term on the left has an extra a because on the right there is an a being multiplied to the trinomial that needs to be accounted for. Simplify by factoring the trinomial that is now a perfect square and solve for y.
Graphing
The graph of a quadratic function is called a parabola. If a>0 then the parabola opens at the top, otherwise it opens at the bottom. Often, the easiest way to graph a parabola is to identify its vertex, x-intercept(s), and the y-intercept then use those to draw a curve.
The x-intercepts can be found by factoring or with the quadratic formula. The quadratic formula is an extremely helpful equation to solve for the roots of a quadratic in the form y= ax2+bx+c. The quadratic formula says:
The quadratic formula is also helpful to identify the number and types of intercepts for a parabola. If we break it up into parts, we can see that 
. The first part of this represents the center of the parabola. Then we go a distance of 
in either direction to get the intercepts, because a parabola is symmetrical. So, the number of solutions is determined by the square root term.
b2-4ac is often called the Discriminant and can be labelled as D. There are three cases.
Case 1: D>0 There are two real solutions to the square root. Thus, there are two x-intercepts.
Case 2: D=0 There is one real solution (that occurs twice, called a “double root”). Thus, there is one x-intercept and it is the same as the vertex.
Case 3: D<0 There are no real solutions, but there are two complex solutions. The solutions are called complex because there is a square root with a negative number inside, so we can think of these as our ‘imaginary’ solutions. Thus, there are no x-intercepts.
Each of these cases are pictured below, in order:
Example 4: Find the x-intercepts of y=3x2-4x+7
Solution: Generally, the first method is to factor. However, on quick inspection we should be able to identify that this will be very difficult to factor, if it can at all factor. So, we should use the quadratic formula.
This quadratic has no real roots, but instead has two complex roots.
Example 5: Graph the equation y= –x2+6x-2
Solution: We can begin by finding the vertex, x-intercepts, y-intercept and then use those points to draw the parabola.
Domain
Quadratic equations, like linear equations, have no domain restrictions. The domain of a quadratic is always D: x∈
.
Range
Since a quadratic ‘opens upwards’ or ‘opens downwards’ there is a restriction on the range. To find the range of a quadratic we should start by identifying the maximum or minimum value of the function. This occurs at the vertex of the parabola which is at the y-value 
. Another way to find the vertex is to locate the x-coordinate of the vertex using XXXX and plugging that value into the function to find the y-coordinate. If a>0 then it will have a minimum because it opens upwards, otherwise it will have a maximum. If the value is a minimum, the range is y∈[minimum,∞). If the value is a maximum, the range is y∈(-∞,maximum].
Example 6: Find the range of y=2x(x-3)
Solution: It is helpful to put this into either vertex or standard form. For this equation, distributing is enough to make it match standard form: y=2x2-6x
Now we can find the x-coordinate of the vertex, which helps us to find the y-coordinate.
Since a=2>0 this is a minimum value, and the range of the function is y ∈ [-9/2, ∞).
Applications
Quadratic functions are particularly helpful in solving problems associated with gravity. The reason for this is that gravity is measured in m/s2 so in physics we can create a formula:
If we rearrange this, we can say:
This looks an awful lot like a quadratic equation because it is! In this case, t is the input variable and yf is the output variable. So, this is a quadratic function in terms of t. The a in this equation represents the acceleration due to gravity.
Example 7: Find the maximum height for an object thrown given it takes 7 seconds to hit the ground and gravity is measured at -9.8m/s2.
Solution: First we should use plug in the given information to the second equation with ∆y=0 and h(t)=0 because we assume the object starts and lands on the ground.
Since we know the object takes 7 seconds to hit the ground, we know that (t-7) must be one of the roots.
We can rewrite the equation so h(t)=-4.9t2+34.3t and use the vertex equation to find the maximum: 
So, the maximum height is 60.025 meters.
Example 8: How long does it take an object to land if it is initially thrown at a vertical speed of 37meters per second from 2 meters above ground?
Solution: Using h(t)=-4.9t2+37t+2
We use the quadratic formula to find the roots:
The t value close to 0 is negative, and happens before it is thrown, so we should ignore that value, so t=7.6046 seconds is the time at which the object hits the ground.
Review
A quadratic equation is a polynomial where the highest exponent is two. That is to say, there is an x2 term and any other x term present has an exponent of 1 or 0.
Domain: D: x∈.
Range: y ∈[minimum,∞) or y∈(-∞,maximum]
Maximum and Minimum defined as f(xvertex ) where xvertex=-b/2a or y=(4ac–b2)/4a
Graph appears as a ∪ or ∩ shape depending on leading coefficient.
Equations
