AP Precal - FRQ Model Solutions

vertical asymptote (rational functions)

Because -3 and 3 are the real zeroes of the polynomial in the denominator of t and not real zeros of the polynomial in the numerator of t, the graph of t has vertical asymptotes at x=-3 and x=3.

zeroes (rational functions)

The real zeroes of t correspond to the real zeroes of the numerator for values in the domain of t. The real zeroes are x=-2 and x=4. Because the graph of t has a hole at x=1, x=1 is not in the domain of t.

horizontal asymptote (rational functions)

The degree of the polynomial in the numerator of t is the same as the degree of the polynomial in the denominator of t. Therefore, for input values of large magnitude, the quotient of the leading terms is a constant. That constant indicates the location of a horizontal asymptote. 1x³/5x³ = 1/5; y = 1/5 is a horizontal asymptote at the graph of t.

minima/maxima (rational functions)

Where f switches from decreasing to increasing is the location of local minimum output value. Where f switches from increasing to decreasing is the location of local maximum output value. f has a local minimum output value at the point with coordinates (0,214, -41,044) and f has a local maximum at the point with coordinates (3.012, 20.303).

minimum/maximum output (polynomial function)

On the closed interval [-0.5, 1.2], relative extrema for f occur at the endpoints of the interval, of where f switches between decreasing and increasing.
f(-0.5) = -3.919

f(1.2) = -5.096

f(0.7369) = -8.832

Therefore, the absolute minimum output value of f is -8.832, which occurs when the input value is x=0.739. The absolute maximum output value of f is -3.919, which occurs when the input value is x=-0.5.

real zeroes (polynomial functions)

The real zeros of f are the real solutions to the equation f(x) =0 (and also the x-intercepts of the graph of f).

f(x) = 2.5x^4+3x³-2.6x²-5.1x-5.6

The solutions to 2.5x^4+3x³-2.6x²-5.1x-5.6 = 0 are x=-1.6 and x=1.4

Therefore, -1.6 and 1.4 are the real zeroes of f.

nonreal zeros (polynomial function)

The degree of the polynomial function f is 4. Therefore, the maximum possible number of distinct zeros of f is 4. Because f has 2 real zeros, f can have at most 2 non-real zeros.

different functions zeros (polynomial functions)

i) (2x+3)(x+2)(x-1)=0

2x+3=0 or x+2=0 or x-1=0

x=-3/2 or x=-2 or x=1

Therefore, all values of x for which g(x)=0 are x -3/2, x=-2, and x=1.

ii) x³+3x²-4x=0

x(x²+3x-4)=0

x(x-1)(x+4)=0

x=0 or x-1=0 or x+4=0

x=0 or x=1 or x=-4

Therefore, all values of x for which h(x)=0 are x=0, x=1, and x=-4.

all values of k(x)>0 (polynomial functions)

3x²-19x-14=0

(3x+2)(x-7)=0

3x+2=0 or x-7=0

x=-2/3 or x=7

These values of x divide the real numbers into three intervals: (neg. infinity, -2/3), (-2/3, 7), and &, infinity). Because k(0)=-14<0, k(x)<0 on (-2/3, 7).

Because k is a quadratic function with 2 x-intercepts and whose graph opens up, k(x)>0 on the other 2 intervals (neg. infinity, -2/3) and (7, infinity).

real and nonreal zeros (polynomial functions)

The real zeros of m include 0 and 5. Because 3-2i is a non-real zero of m, the conjugate 3+2i is also a non-real zero of m. Therefore, the four zeros of m are 0, 5, 3-2i, and 3+2i. Expression for m(x): m(x) = x(x-5)(x²-6x+13)