Question

1. let $f(x)=\

$$\begin{cases}(x + 9)^2&x < - 8\\\\3x + 4&-8\\leq x < 2\\\\\\sqrt{x - 2}&x\\geq2\\end{cases}$$

$
which of the following statements is true?
a) $f$ is increasing for all $x$
b) $f(0)=81$
c) there is an open circle on the graph of $f$ at $(2,10)$
d) $f(2)$ is undefinded

1. which of the following functions has an axis of symmetry that is parallel to the $y -$

a) $f(x)=x^2-3x + 1$
b) $f(x)=x^3-x$
c) $f(x)=2x - 5$
d) $f(x)=\frac{1}{x}$

1. many large cities have electrically powered scooters for tourists to rent. the tabl

some data about the relationship between the time spent on the scooter and th
renting the scooter. which equation could best be used to model $c(t)$, the cost
scooter for $t$ minutes?
$\

$$\begin{array}{c|c|c|c|c}\\text{time (minutes)}&5&8&14&16\\\\\\hline\\text{cost }(\\$)&\\$1.75&\\$2.20&\\$3.10&\\$3.40\\end{array}$$

$
a) $c(t)=1 + 0.15t$
b) $c(t)=0.35t$
c) $c(t)=0.78\sqrt{t}$
d) $c(t)=0.06t^2-0.63t + 3.4$

Explanation:

Step1: Check each option for question 4

• Option A:

For \(x < - 8\), \(y=(x + 9)^{2}=x^{2}+18x + 81\), its derivative \(y^\prime=2x + 18\). When \(x=-9\), \(y^\prime = 0\). For \(x<-9\), \(y^\prime<0\), the function is decreasing. So \(f\) is not increasing for all \(x\).

• Option B:

Since \(0\) satisfies \(-8\leqslant x<2\), we use \(f(x)=3x + 4\). Then \(f(0)=3\times0 + 4=4
eq81\).

• Option C:

For \(x < 2\), when \(x = 2\), using \(y = 3x+4\), \(y=3\times2 + 4=10\). For \(x\geqslant2\), \(y=\sqrt{x - 2}\), when \(x = 2\), \(y=\sqrt{2-2}=0\). So there is an open - circle at \((2,10)\) on the part of the graph from \(y = 3x + 4\).

• Option D:

For \(x\geqslant2\), when \(x = 2\), \(f(2)=\sqrt{2-2}=0\), so \(f(2)\) is defined.

Step2: Check each option for question 5

• Option A:

For a quadratic function \(y = ax^{2}+bx + c\), the axis of symmetry is \(x=-\frac{b}{2a}\). For \(y=x^{2}-3x + 1\), \(a = 1\), \(b=-3\), the axis of symmetry is \(x=\frac{3}{2}\), which is parallel to the \(y\) - axis.

• Option B:

For \(y=x^{3}-x\), \(y^\prime=3x^{2}-1\), it is a cubic function. The graph of a non - constant cubic function does not have an axis of symmetry parallel to the \(y\) - axis.

• Option C:

\(y = 2x-5\) is a linear function (a straight line), and it does not have an axis of symmetry parallel to the \(y\) - axis.

• Option D:

\(y=\frac{1}{x}\) is a hyperbola, and its axis of symmetry is \(y = x\) and \(y=-x\), not parallel to the \(y\) - axis.

Step3: Check each option for question 6

• Option A:

When \(t = 5\), \(C(5)=1+0.15\times5=1 + 0.75=\$1.75\); when LXI1, \(C(8)=1+0.15\times8=1 + 1.2=\$2.2\); when \(t = 14\), \(C(14)=1+0.15\times14=1+2.1=\$3.1\); when LXI3, \(C(16)=1+0.15\times16=1 + 2.4=\$3.4\). All the values match the table.

• Option B:

When \(t = 5\), \(C(5)=0.35\times5=\$1.75\), when LXI1, \(C(8)=0.35\times8=\$2.8
eq\$2.2\).

• Option C:

When \(t = 5\), \(C(5)=0.78\sqrt{5}\approx0.78\times2.24\approx\$1.75\), when LXI1, \(C(8)=0.78\sqrt{8}\approx0.78\times2.83\approx\$2.21\approx\$2.2\), when LXI2, \(C(14)=0.78\sqrt{14}\approx0.78\times3.74\approx\$2.92
eq\$3.1\).

• Option D:

When \(t = 5\), \(C(5)=0.06\times5^{2}-0.63\times5 + 3.4=0.06\times25-3.15 + 3.4=1.5-3.15 + 3.4=\$1.75\), when LXI1, \(C(8)=0.06\times8^{2}-0.63\times8 + 3.4=0.06\times64-5.04 + 3.4=3.84-5.04 + 3.4=\$2.2\), when \(t = 14\), \(C(14)=0.06\times14^{2}-0.63\times14 + 3.4=0.06\times196-8.82 + 3.4=11.76-8.82 + 3.4=\$6.34
eq\$3.1\).

Answer:

1. C. There is an open circle on the graph of \(f\) at \((2,10)\)
2. A. \(f(x)=x^{2}-3x + 1\)
3. A. \(C(t)=1 + 0.15t\)