Fun fact about square root functions:
The word quadratic comes from the word 'quad' which means square. Which is why the equation for a square root function is y=x^2. The standard for of this function is y = ax^2 + bx +c The graph below this shows how manipulating this equation can shift the graph. |
History of the Quadratic Function:
The quadratic function was first created in 2000 B.C.by Babylonians to solve quadratics in radicals. When they solved quadratics they would focus on finding a length, and the length that was found was the root of the quadratic equation. The first steps towards finding a general formula to solve quadratic equations trace back around 300 BC, mathematicians Pythagoras and Euclid found a method to solve the quadratic equation by focusing strictly on geometry. Pythagoras noted that the ratios found a general procedure to solve the quadratic equation. During Euclid's time, there didn't exist the same notation for numbers and formulas that exist today, and because of that, it was not possible to compute the square root of any number by hand. In 800 AD, al-Khwarizmi, a Persian mathematician, created six new equations involving roots, squares of roots, and numbers which he used to solve quadratic equations, and derived the quadratic equation. The algebra used by him was entirely rhetorical, and he rejected negative solutions. These discoveries led Abraham bar Hiyya Ha-Nasi to publish a book with the complete solution of the quadratic formula. In the year 1545, Gerolamo Cardano, a mathematician from Italy and also one of the best algebraists of his time, collected all the works related to equations. Later in the year 1594, the quadratic formula was fist obtained by Simon Stevin, who was born in Bruges (now known as Belgium). |
Quadratic Functions in the Real World"In athletic events that involve throwing things, quadratic equations are highly useful. Say, for example, you want to throw a ball into the air and have your friend catch it, but you want to give her the precise time it will take the ball to arrive. To do this, you would use the velocity equation, which calculates the height of the ball based on a parabolic (quadratic) equation. So, say you begin by throwing the ball at 3 meters, where your hands are. Also assume that you can throw the ball upward at 14 meters per second, and that the earth's gravity is reducing the ball's speed at a rate of 5 meters per second squared. This means that we can calculate the height, using the variable t for time, in the form of h = 3 + 14t - 5t^2. If your friend's hands are also at 3 meters in height, how many seconds will it take the ball to reach her? To answer this, set the equation equal to 3 = h, and solve for t. The answer is approximately 2.8 seconds."
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Quadratic Function Graph and Equation
The graph of a quadratic function makes a shape known as a parabola.The parent graph is shown above, as y=x^2.The standard form equation is
y=ax^2 + bx + c. There is also vertex form where y=a(x-h)^2 +k Parabolas can be altered from the parent graph (shown above) by being vertically or horizontally stretched or compressed, horizontally or vertically shifted, and horizontally or vertically reflected. The domain of almost every quadratic function is going to be all real numbers, because all x-value inputs are valid. The range of the function is those points greater than of less than the vertex of the graph. For example, if the graph vertex is (0,3) and it is a positive graph, the range would be all numbers greater than 3. |
Common Core StandardsNorth Carolina Common Core says that students will:
Construct and compare linear, quadratic, and exponential models and solve problems. This standard will help students use different types of problems with different types of functions. To learn more about the common core standards in Nroth Carolina, click the link below. |