如果 x 增加了 25%，求 y 的百分比变化。 y = $\dfrac{k}{x^2}$，其中 k 为常数。 答案 ... % 已知当 x = $\dfrac{1}{2}$ 时，y = 16，求 连接 y 和 x 的方程。 答案 ... 当 y = 100 时，x 的值。 答案 x = ... 画出表示 y 和 x 之间关系的图像草图。

1. Find the new value of $x$ as a multiplier of the original value.

Increasing $x$ by $25\%$ gives
$x_{\text{new}}=125\%\text{ of }x=1.25x=\frac{5}{4}x.$

2. Use $y=\dfrac{k}{x^2}$ to find the percentage change in $y$.

Since
$y=\frac{k}{x^2},$
the new value of $y$ is
$y_{\text{new}}=\frac{k}{\left(\frac{5}{4}x\right)^2}=\frac{k}{\frac{25}{16}x^2}=\frac{16}{25}\cdot\frac{k}{x^2}=\frac{16}{25}y.$
So $y$ becomes $\frac{16}{25}=0.64=64\%$ of its original value.

Percentage change in $y$:
$64\%-100\%=-36\%.$

1. Substitute the given values into the given relation.

Given $y=16$ when $x=\dfrac{1}{2}$,

$16=\dfrac{k}{\left(\dfrac{1}{2}\right)^2}$

2. Solve for $k$ and write the equation connecting $y$ and $x$.

$16=\dfrac{k}{\frac14}$

$16=4k$

$k=4$

Therefore,

$y=\dfrac{4}{x^2}$

1. Substitute $y=100$ into the equation from part (b)(i).

From part (b)(i),
$y=\frac{4}{x^2}$
When $y=100$,
$100=\frac{4}{x^2}$

2. Solve the equation for $x$.

$100x^2=4$
$x^2=\frac{4}{100}=\frac{1}{25}$
Since $x>0$,
$x=\sqrt{\frac{1}{25}}=\frac{1}{5}$

1. Sketch the inverse-square graph with the correct shape and domain.

Since $y=\dfrac{4}{x^2}$ and $x>0$, the graph lies only in the first quadrant. It is a smooth decreasing curve, close to the $y$-axis when $x$ is small and approaching the $x$-axis as $x$ becomes large.