Estimando possiveis soluções

To estimate a possible solution to an equation, you can always look at the graph. We can estimate different features in order to create the equation of the function.

Let's take the equation $y=x^2$. Here's what the graph looks like.

Okay so what if I asked you to find the value of $y=x^2$ when $x=2.7824$? Well I certainly can't do that in my head, and I don't have a calculator near me. But I can estimate the value by looking at the graph. We are essentially going to find the corresponding $y$ value to our $x$ value by reading the graph!

This looks like a quadratic function. First let's find $x=2.7824$ on the $x$-axis.

Here's roughly where $x=2.7824$ is on the $x$-axis. Now we have to move up in a straight line until we reach our graph.

And now we just have to move in a straight line towards the $y$-axis and see what number it lands on.

As you see it lands around $7.5$. So we can say that when $y=x^2$ where $x=2.7824$ that $y\approx7.5$. If you want to be a little more accurate you can notice $x=2.7824$ is a little above $x=2.75$. So you can choose a $y$ value a little above $7.5$. Let's say $7.68$.

And hey look I found my calculator. Let's see what the actual answer is and see how close we were. If $y=x^2$ where $x=2.7824$ then $y=7.74174976$. As you can see both $y=7.5$ and $y=7.68$ were fairly close.

Be careful though. While estimations might be helpful for some quick calculations, it's always best to be as accurate with your numbers as much as possible. This method is good for approximations, or if you're looking for a way to double check your work, but it does not provide an accurate number. Even if the value you found is right on a line, it is always good practice to double check with exact calculations.

Functions describe the relationship between unknowns. For example: $$y=x+3$$ This is a function, it describes the relationship between $x$ and $y$. We can say that three more than $x$ is $y$!

We can plot, or graph, functions to help up visualize the functions, which in turn helps us estimate possible solutions.

What is $y$ when $x$ is $9$? We can calculate it, but we can also graph it and estimate the solution.

Find the point on the function (in red) where $x = 9$, and estimate what $y$ value it corresponds to.

Draw a line from the point on the line to the corresponding $y$ value. Make sure your line is straight!

In this case, $y=12$.

This is a very quick and simplified explanation of functions and graphing functions. You can learn more in depth knowledge on functions and graphing functions.

(Video) Finding function values given a graph

by Bethany M.

In this video, Bethany discusses how to find values or solutions of a graph given and $x$ value.

$\large{\text{Summary}}$

The steps to finding these solutions are:

1. Find the $x$-value on the graph.
2. Go up or down on the $x$-value until you hit the graphed function.
3. Go across to the $y$-axis to find the $y$-value that corresponds with the $x$-value that is given.

First, we find $f(1)$. By following those steps, we see that $$f(x)=9$$

Next, we find $f(-4)$. Again, following the steps, we see that $$f(x)=-4$$

We also see that these solutions or $y$-values that correspond to $x$ are the same as ordered pairs on the graph.

Lastly, when we find $f(-7)$, we see that there is a hole in the graph. So, the solution there is undefined.

Look at the rest of the video to work through a harder example.

We can also estimate a value or solution just looking at the equation. Say we have the equation $12=6.023842x+2.34150$. What is the estimated solution for $x$?

To do this, we can simplify $6.023842$ to $6$ and $2.34150$ to $2$. $$\begin{align} 12&=6.023842x+2.34150\\ 12&\approx6x+2\\ 12\,\color{#8925AE}{-\,2}&\approx6x+2\,\color{#8925AE}{-\,2}\\ 10&\approx6x\\ \frac{10}{\color{#29A603}{6}}&\approx\frac{6x}{\color{#29A603}{6}}\\ \frac53&\approx x\\ x&\approx1.5 \end{align}$$

If we wanted the most accurate value, we would leave $6.023842$ as the slope and $2.34150$ as the $y$-intercept, but simplification makes the equation easier to handle.

Solutions to Equations with Graphing

When a variable is solved for in an equation, the solved value of the variable is what we call a solution. A solution in math satisfies the equation, therefore making the equation true. Often times, the solution of an equation is where the graph crosses the $x$-axis.

Example $1$

What is the solution of $x=4+5$?

To find this solution, we would just add $4+5$ to get $x=9$. We can graph this to visualize which $x$ values satisfy the equation.

Example $2$

What is the solution to $y=x+2$?

We would plug in $0$ for $y$ and solve for $x$. This gives us $0 = x +2$, which we can solve to get a solution of $x = -2$. Look at the graph below and notice that the graph crosses the $x$-axis at $x=-2$.

Example $3$

What is the solution to $y=\sqrt{x}$?

To find this out we can plug in $0$ for $y$ or look at the graph. We see that the graph crosses the $x$-axis at $x=0$.

Solutions to Inequalities

When a variable is solved for in an inequality, we also call that a solution. Remember, a solution in math satisfies the equation. When discussing inequalities, there can be multiple solutions that satisfy the equation.

Example $1$

What is the solution of $x + 4<5$?

To find this solution, we first want to isolate the $x$ variable. We can successfully do this by subtracting $4$ from both sides to isolate the variable $x$ to the left side of the equation (because of the Addition and Subtraction Property of Equality). $$\begin{align} x + 4 &< 5\\ x + 4 \,{\color{#8925AE}{-\,4}} &< 5 \,{\color{#8925AE}{-\,4}}\\ x &< 1 \end{align}$$ We can graph this to visualize which $x$ values satisfy the equation.