Chapter 1.2, Problem 88AYU

Make up an equation with the intercepts $\left(2,0\right)$, $\left(4,0\right)$ and $\left(0,1\right)$. Compare your equation with a friend’s equation. Comment on any similarities.

To determine

An equation with the intercepts $\left(2,0\right),\left(4,0\right)$ and $\left(0,1\right)$.

Answer to Problem 88AYU

Solution:

An equation with the intercepts $\left(2,0\right),\left(4,0\right)$ and $\left(0,1\right)$ is $y=\frac{1}{8}\left(x-2\right)\left(x-4\right)$.

Explanation of Solution

Given Information:

Intercepts $\left(2,0\right),\left(4,0\right)$ and $\left(0,1\right)$.

Explanation:

Let the intercept points be $\left(2,0\right),\left(4,0\right)$ and $\left(0,1\right)$.

Now, to find the equation $y=f\left(x\right)$ by using the intercepts $\left(2,0\right),\left(4,0\right)$ and $\left(0,1\right)$.

The points, if any, at which a graph crosses or touches the coordinate axes are called the intercepts of the graph.

That is, the $x-$ intercept of the graph is a point where $y=0$, and the $y-$ intercept of the graph is a point where $x=0$.

For the points $\left(2,0\right)$ and $\left(4,0\right)$, the $y$-coordinates equal to 0.

Thus, $x=2$ and $x=4$ are $x-$ intercepts.

Then, $\left(x-2\right)$ and $\left(x-4\right)$ are factors of the equation $f\left(x\right)$.

For the points $\left(0,1\right)$, the $x$-coordinates is 0.

Thus, $y=1$ is $y-$ intercept.

Therefore, the function would be $y=f\left(x\right)=A\left(x-2\right)\left(x-4\right)$, where $A$ is leading coefficient.

To find $A$, use $y-$ intercept.

It gives $f\left(0\right)=1$

Plug in $x=0$ in $y=f\left(x\right)=A\left(x-2\right)\left(x-4\right)$

$\Rightarrow f\left(0\right)=A\left(0-2\right)\left(0-4\right)$

Since $f\left(0\right)=1$,

$\Rightarrow 1=8A$

$\Rightarrow A=\frac{1}{8}$

Hence, the required equation with the intercepts $\left(2,0\right),\left(4,0\right)$ and $\left(0,1\right)$ is $y=\frac{1}{8}\left(x-2\right)\left(x-4\right)$.