4.4 Quadratic Functions
The formula:
gives us the general equation for a quadratic function, also known as a parabola.
We've now explored how the individual components of f(x)=-4.9x^{2}+2.45x+1.7 transform the square function into different versions of the same picture; though the constants may flip, stretch, or translate the square function, in each case, we've discovered that they all looked basically the same. This is a very powerful observation! As mathematicians, it tells us that we can study a general function that looks like:
f(x)=ax^2+bx+c, a\ne0
where a, b, and c are constants, (a≠0 or we'd just have a linear function) and apply our understanding of this to the infinite variety of specific functions that share this pattern. A mathematical creature of this form is known as a quadratic function or a parabola.
Our function f(x)=-4.9x^{2}+2.45x+1.7 is then just a quadratic function with coefficients
a = -4.9, b = 2.45, c = 1.7. Since the "a" term, also known as the leading coefficient, is negative, we can already see that our quadratic will be upside down. This fits our intuition, because when you toss an object, it travels in an upside-down arc. The 1.7 tells us that our function will shift our function up a height of 1.7 meters. This makes sense if 1.7 meters (5' 7") is the initial height at which you toss the egg.
With a deeper understanding of what quadratic functions look like, let's revisit the algebraic questions that we'd like to answer. We've spent a lot of time just "getting to know" quadratics and it may seem that we've taken a bit of a detour. However, being able to picture the mathematical creature that you're studying almost always gives you additional insight into how it behaves algebraically.
- When does the egg reach a height of 1.7 meters?
- When does the egg hit the ground?
- When does the egg reach its maximum height?
To find out when the egg is at height 1.7 meters, we're actually trying to solve the equation
\displaystyle -4.9x^{2}+2.45x+1.7=
To determine when the egg reaches its maximum height, we'd like to know at what time it's at the "hump" (formally known as the vertex of a parabola). Since our parabola is upside down, the x-coordinate of the vertex will then give us the time that the parabola is at the highest point.
Finally, when we ask "When does the egg hit the ground?", we're actually trying to solve the equation
\displaystyle -4.9x^{2}+2.45x+1.7=
Armed with our new understanding of what parabolas look like, we'll answer these questions after developing some new algebraic tools. Let's first try to figure out when the egg is at a height of 1.7 meters.
