ACCUPLACER Next Generation Quantitative Reasoning Algebra and Statistics Version 3 Questions

Explanation

To find the value of a, we can use the points provided in the table and the given function f(x)=kax. First, use the point (0,1/2) since any base raised to the power of 0 is 1. Substituting these values into the function gives 1/2=ka^0, which simplifies to 1/2=k(1), so k=1/2. Now that we have the value of k, we can substitute it back into the function to get f(x)=1/2ax. Next, use another point from the table, like (1,2), to solve for a. Substituting the values into the equation gives 2=1/2a^1. To isolate a, multiply both sides by 2, resulting in a=4. This value can be confirmed with the remaining point (−1,1/8), where 1/2(4)^-1=1/2(1/4)=1/8, which is consistent. The other options are incorrect as they do not satisfy the function with the given points.

Explanation

The condition f(x)=f(−x) defines an even function, which is a function that is symmetric with respect to the y-axis. To check if a function is even, one must substitute −x for x and see if the resulting expression is the same as the original function. When we test option D, f(x)=(x^3−1)^3, and substitute −x, we get f(−x)=((−x)^3−1)^2=(−x^3−1)^2. Expanding this results in (x^3+1)^2, which is not equivalent to the original function (x^3−1)^2. In contrast, options A, B, C, and E all satisfy the condition because any term with an even exponent remains unchanged when −x is substituted for x, making them even functions

Explanation

To solve the expression, first replace each trigonometric function with its value at 45 degrees. The sine of 45 degrees is the square root of 2 divided by 2. The cosine of 45 degrees is also the square root of 2 divided by 2. The tangent of 45 degrees is 1. When you multiply the values, you get (the square root of 2 / 2) multiplied by (the square root of 2 / 2) multiplied by 1. The product of the two square roots of 2 is 2, and the product of the two denominators is 4. This results in the fraction 2/4, which simplifies to 1/2.

Explanation

An expression that can be factored into the form of a squared binomial is called a perfect square trinomial. This type of expression always has three terms: the square of a first term, a middle term that is twice the product of the two terms in the binomial, and the square of a second term. The expression 4x^2+20xy+25y^2 fits this exact pattern. The first term, 4x^2, is the square of 2x, and the last term, 25y^2, is the square of 5y. The middle term, 20xy, is exactly twice the product of 2x and 5y. The other expressions do not fit this pattern; the first is a difference of two squares, the second is a sum of two squares, and the last has a negative term where a positive term is required for it to be a perfect square trinomial.

Explanation

The original graph of y=5x^2−20x+17 is a parabola that opens upwards and has its vertex at a y-value of -3. For the new parabola, y=5x^2−20x+(17+k), to not intersect the x-axis, its vertex must be shifted upwards so that its lowest point is above the x-axis. The term plus k represents a vertical shift. Therefore, the y-value of the new vertex, which is the original y-value plus k, must be greater than 0. The original y-value is -3, so we need to find a value for k such that −3+k>0. By adding 3 to both sides of the inequality, we find that k must be greater than 3. The only answer choice that satisfies this condition is 4.