Since the shown parabola opens upward, the coefficient of $x^2$ in the equation $y = ax^2 + c$ must be positive. Given that $a$ is positive, $-a$ is negative, and therefore the graph of the equation $y = -a(x-b)^2 + c$ will be a parabola that opens downward. The vertex of this parabola is $(b, c)$, because the maximum value of $y$, $c$, is reached when $x = b$. Therefore, the answer must be choice B.

First, we must note that the parabola is given in vertex form $y=a(x-h)^2+k$. So we know the vertex is $(h,k)=(-1, 3)$. Now we need to find the $y$-intercept. Since the $y$ - intercept will always be where $x=0$ we find that $y=5$ therefore $y$ - intercept is $(0,5)$. All we need to do now is calculate the slope between points $(-1,3)$ and (0,5)$ which is $m=\frac{5-3}{0-(-1)}=\frac{2}{1}=2$

Note that this quadratic function is already in factored form $y=a(x-x_1)(x-x_2)$. The $y$ coordinate of vertex of a quadratic function in the factored form is $f\Big(\frac{x_1+x_2}{2}\Big)$ therefore $k=f\Big(\frac{-2+10}{2}\Big)= f(4)= \frac{1}{2}(4+2)(4-10)=-18$

Exponential growth/decay functions model quantities that increase or decrease by a fixed percent during each time period $y=ab^t=a(1\pm r)^t, a>0$ where $a$ is initial amount, $b$ the growth (or decay factor) is the ratio between two consecutive $y$-value and $y$ is amount after $t$ time periods. In this problem the equation looks like $y=10(1.20)^t$ but notice that we have to change the exponent $t$ into $4t$ since the exponential growth happens four times so equation become $y=10(1.20)^{4t}$.

Exponential growth/decay functions model quantities that increase or decrease by a fixed percent during each time period $y=ab^t=a(1\pm r)^t, a>0$ where $a$ is initial amount, $b$ the growth (or decay factor) is the ratio between two consecutive $y$-value and $y$ is amount after $t$ time periods. In this problem the equation looks like $y=250000(1.30)^t$ but notice that we have to change the exponent since the exponential growth happens only every 20 years, so equation become $y=250000(1.30)^{\frac{t}{15}}$

Since the function $h$ is exponential, it can be written as $h(x) = ab^x$, where $a$ is the $y$-coordinate of the $y$-intercept and $b$ is the growth rate. Since it's given that the $y$-coordinate of the $y$-intercept is $d$, the exponential function can be written as $h(x) = db^x$. These conditions are only met by the equation in choice D.

$n^4=20x$
$n^4=20n^3$
$n^4-20n^3=0$
$n^3(n-20)=0$ therefore $n=20$ and $x=8000$

For $x>1$ and $y>1, x^{\frac{1}{3}}$ and $y^{\frac{1}{2}}$ are equivalent to $\sqrt[3]{x}$ and $\sqrt{y}$ respectively. Also, $x^{-2}$ and $y^{-1}$ are equivalent to $\displaystyle \frac{1}{x^2}$ and $\displaystyle \frac{1}{y}$, respectively. Using these equivalences, the given expression can be rewritten as $\displaystyle \frac{y\sqrt{y}}{x^2\sqrt[3]{x}}$

Write an equation which describe this situation $7x=24+x$. Solving for $x$ we get $7x-x=24$ or $6x=24$ so $x=4$. Therefore $x^2=16$ and $\sqrt{x}=2$ so $x^2$ is 14 more than $\sqrt{x}$

Square both sides of $2x-5 =\sqrt{9-40x}$ to reach $4x^2- 20x + 25 = 9-40x$. This simplifies to $4x^2 + 20x + 16 = 0 \Rightarrow x^2 + 5x + 4 = 0 \Rightarrow (x + 4)(x+ 1) = 0$.\\ Therefore all possible roots are $x = -1,-4$. However, when substituted into the original equation, the left hand side is negative. Assuming the principal square root, no working solution would allow the left side to be negative. Therefore, there are no values of $x$ which make the equation true.

Correct Answer is A. $\displaystyle 1+\frac{\frac{1}{x}}{\frac{1}{y}}=1+\frac{1}{x}\cdot \frac{y}{1}=1+\frac{y}{x}=\frac{x+y}{x}$

Correct Answer is: B. Average of $x$ and $\frac{1}{x}$ is $\displaystyle \frac{x+\frac{1}{x}}{2}=\frac{\frac{x^2}{x}+\frac{1}{x}}{2}=\frac{\frac{x^2+1}{x}}{2}=\frac{x^2+1}{x}\cdot \frac{1}{2}=\frac{x^2+1}{2x}$