If the inequality y> x^2 - 7x + 12 is graphed, is the point (5, 2) in the shaded region?
The SAT is all about time management. The quickest way to do this problem is by simply plugging in the point that was given. Use 2 as the y-value and 5 as the x-value and you will get 2>2. Since this is false, the point would not be in the shaded region, which represents all solutions to the inequality. The lengthier method is graphing. By factoring the expression x^2 - 7x + 12 into (x-3)(x-4), we know that the parabola has two real roots at x=3 and x=4. By sketching this graph and shading the area inside the parabola, we can see that this point is on the parabola but not in the shaded region. The inequality says greater than, not greater than or equal to, so the points on the parabola are not solutions to the inequality.
If the period of a standard sine function y=sin(x) is doubled, how are the amplitude, zeroes, and frequency affected?
The period of a sine function is the distance traveled on the x-axis as the function cycles or repeats its shape. The period for y=sin(x) is 2pi. As the period is doubled to 4pi, picture the graph being horizontally stretched so that one cycle takes up double the amount of room. If it is being stretched lengthwise, the vertical distance traveled is not affected, so the amplitude remains the same. The zeroes or roots of this function will not be at the same points they were originally at but will be farther apart now. To find the new zeroes, simply double the x-value of the old zeroes. Lastly, the frequency is always the reciprocal of the period, so 1/4pi instead of 1/2pi. The period increased, so the frequency decreased.
What is the difference between a line that is tangent to a circle and a line that is secant to a circle? Which of the two could possibly show the diameter of the circle?
A tangent touches the circle at just one point and creates a right angle with a radius of the circle. Due to the infinite number of points on a circle, there is an infinite number of possible tangents. There is also an infinite number of secants, but a secant must touch exactly two points on the circle so that one secant to the circle will run through the diameter or the longest chord in a circle. Tangents can never be the diameter of the circle because they do not touch two sides of the circle.