2.1 KiB
Notes
To convert an equation in the form of y = (x^2-3x) into a square equivalent, you:
- Take the second value,
3, then half it, giving you (\frac{3}{2}). - Then to rebalance the equation, you're going to square that value (giving you
\frac{9}{4}), then add it to the other side, multiplying it byaif necessary, whereais what the parentheses are multiplied by in the forma(x - h)^2. This will give you an equation that looks something like:\frac{9}{4} + y = (x -\frac{3}{2})^2. - Finally, you can rebalance the equation by subtracting
\frac{9}{4}from both sides, giving youy = (x - \frac{3}{2})^2 - \frac{9}{4}. This equation should be equal to the original.
y = -5x^2 -20x + 13 Given the above equation, you can factor out a -5, resulting in the equation -5(x^2+4x) + 13). Half of 4 is 2, and because the inside is multiplied by -5, -5 *4 = -20, so you add -20 to the other side to equalize the equation, resulting in an equation in the form of -20 + y = -5(x+2)^2+ 13. This simplifies down to y = -5(x+2)^2 + 33.
Forms
Standard form (vertex form)
y = a(x - h)^2 + k To convert to standard form given a vertex of a quadratic equation and a point that falls along that line, plug values in for everything and solve for a.
Quadratic form
y = a^2 + bx + c End Behavior of functions
If the largest exponent of a function is even, both sides of a function will point the same way.
Given there's no negative coefficient, as x goes towards infinity, f(x) will go towards infinity. As f(x) goes towards -\infty, f(x) will still go to infinity. A negative coefficient will flip this.
x \rightarrow \infty, \space f(x) \rightarrow \infty x \rightarrow -\infty, \space f(x) \rightarrow -\infty If the largest exponent of a function is odd, each side of the function will point towards a different direction. Given there's no negative coefficient, the left side of the graph will point down, and the right side will point up. A negative coefficient will flip this.
The least degree of a polynomial is the number of turning points + 1.