Math  /  Algebra

QuestionWrite an equation for the linear function f satisfying the conditions below. Then, graph y=f(x)y=f(x). f(5)=2 and f(1)=3f(-5)=-2 \text { and } f(1)=3
Write the equation y=f(x)y=f(x). y=56x+136y=\frac{5}{6} x+\frac{13}{6} (Type an equation. Use integers or fractions for any numbers in the equation.) Use the graphing tool to graph the linear equation.

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a line that goes through the points (5,2)(-5, -2) and (1,3)(1, 3), and then graph it! Watch out! Don't mix up the xx and yy coordinates!
Also, double-check your calculations to avoid simple arithmetic errors.

STEP 2

1. Find the slope.
2. Find the y-intercept.
3. Write the equation.
4. Graph the line.

STEP 3

Let's **start** by finding the **slope** of our line.
Remember, the slope is the "rise over run," or how much the yy value changes for every change in the xx value.
The formula for the slope mm between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is: m=y2y1x2x1m = \frac{y_2 - y_1}{x_2 - x_1}

STEP 4

In our case, we have the points (5,2)(-5, -2) and (1,3)(1, 3).
Let's **plug** these values into our **slope formula**: m=3(2)1(5)m = \frac{3 - (-2)}{1 - (-5)}

STEP 5

Now, let's **simplify**: m=3+21+5=56m = \frac{3 + 2}{1 + 5} = \frac{5}{6} So, our **slope** is m=56m = \frac{5}{6}.
Awesome!

STEP 6

Now that we have our **slope**, we can use the **point-slope form** of a linear equation, which is: yy1=m(xx1)y - y_1 = m(x - x_1) We can use either of our points; let's use (1,3)(1, 3) because the numbers are simple!

STEP 7

**Substitute** the **slope** m=56m = \frac{5}{6} and the point (1,3)(1, 3) into the **point-slope form**: y3=56(x1)y - 3 = \frac{5}{6}(x - 1)

STEP 8

Now, let's **solve for** yy to get the **slope-intercept form** (y=mx+by = mx + b), which will give us our **y-intercept** (bb): y3=56x56y - 3 = \frac{5}{6}x - \frac{5}{6} y=56x56+3y = \frac{5}{6}x - \frac{5}{6} + 3To add 33 to 56-\frac{5}{6}, let's rewrite 33 as 186\frac{18}{6}: y=56x56+186y = \frac{5}{6}x - \frac{5}{6} + \frac{18}{6} y=56x+136y = \frac{5}{6}x + \frac{13}{6}Our **y-intercept** is 136\frac{13}{6}!

STEP 9

We already found the equation in slope-intercept form while finding the y-intercept!
It's: y=56x+136y = \frac{5}{6}x + \frac{13}{6} So, f(x)=56x+136f(x) = \frac{5}{6}x + \frac{13}{6}.

STEP 10

To graph the line, we can use the two points we were given, (5,2)(-5, -2) and (1,3)(1, 3).
Plot these points on a graph and draw a straight line through them.
You can also use the y-intercept (0,136)\left(0, \frac{13}{6}\right) as another point to help you draw the line accurately.

STEP 11

The equation of the linear function is y=56x+136y = \frac{5}{6}x + \frac{13}{6}.
The graph of the line passes through the points (5,2)(-5, -2) and (1,3)(1, 3), and has a y-intercept at (0,136)\left(0, \frac{13}{6}\right).

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