- A quadratic function is an equation with the independent variable taken to the second degree. the most common form is y=ax2+bx+c for example, y=3×2-2x+7
- For the quadratic function where a=1, b=0 and c=0, the function simplifies to the basic form of y=x
3. Keeping b and c constant, the graph changes when the value for a is changed.
a. a<0, the negative value of a caused the parabola to open down. As a result, there is a maximum value.
b. a>0, the positive value of a caused the parabola to open up. As a result, there is a minimum value.
c. -1<a<1, the value of a caused the parabola to have a wider shape. If the value is positive, the curve opens up; if the value is negative, the curve opens down.
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d. a>1 or a<-1, the value of a caused the parabola to be stretched out, so that the overall shape is much taller and skinnier. If the value is positive, the curve opens up; if the value is negative, the curve opens down.
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4. The maximum or minimum point of a quadratic function is called the vertex. There is a direct relationship between the a value and the opening direction of the curve, as well as the maximum and minimum value.
- When the a value is positive, the parabola opens up, resulting in a minimum value. It is not possible to go below the minimum value.
- When the a value is negative, the parabola opens down, resulting in a maximum value. It is not possible to go above the maximum value.
5. Keeping a (a=1) and c (c=0) constant, the b value changes the graph horizontally and vertically at the same time.
For example, from the basic form of b=0
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when b=10, the graph moved down and left
when b=-0.25, the graph moved up and right
6. Keeping a (a=1) and b (b=0) constant, the c value only changes the graph vertically.
For example, from the basic form of c=0
when c=8, the graph moved up
when c=-8, the graph moved down
Roots are the solutions to the quadratic equation. The roots are found by looking at where the curve crosses the x-axis (x-intercepts).
By adjusting the slider in desmos.com, a quadratic equation can be adjusted so that only a single point touches the x-axis.
For example, with an original equation of a quadratic y=ax2+bx+c, where a=2, b=5 and c=-2, there are two roots. In another word, there are TWO solutions.
By changing the value of c so that c=3.1, the curve is moved up so that there is only 1 root. In this case, there is ONE solution.
When the values are changed to a=2, b=0.1 and c=-0.5, the curve now has TWO solutions at -1 and 1.
To result in a quadratic without any solution, the values are changed to a=2, b=0.1 and c=1. This way, the curve does NOT cross the x-axis and there are NO REAL solutions for this equation.