Find the Function Equation Passing Through Point (7,9): Positive Function Analysis

Q: What does 'positive from the beginning of the axes' actually mean?

This means the function starts at the origin (0,0) and has positive y-values for positive x-values. It's telling you the line passes through the origin!

Q: Why is the answer written as a mixed number instead of an improper fraction?

Both $ \frac{9}{7}x $ and $ 1\frac{2}{7}x $ are correct! Mixed numbers are often preferred because they're easier to visualize - you can see it's slightly more than 1.

Q: How do I convert between mixed numbers and improper fractions?

To convert $ \frac{9}{7} $ to mixed: divide 9 ÷ 7 = 1 remainder 2, so $ 1\frac{2}{7} $. To go back: 1 × 7 + 2 = 9, so $ \frac{9}{7} $.

Q: Could this be a different type of function, not linear?

With only one point given plus the origin constraint, a linear function is the simplest answer. Other functions like $ y = x^2 $ or $ y = \sqrt{x} $ are possible, but linear is most likely intended.

Q: What if I calculated the slope as 7/9 instead of 9/7?

That's backwards! Remember: slope = $ \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} $. From (0,0) to (7,9): rise = 9, run = 7, so slope = $ \frac{9}{7} $.

Linear Functions with Origin Constraints

Given a function that is positive from the beginning of the axes. Plus the point (7,9) on the graph of the function. Find the equation for the function.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the function equations

00:03 Points where the function passes according to the given data

00:09 We'll use the line equations

00:17 We'll substitute the point to find the unknown value B

00:33 This is the value of B

00:41 Now we'll substitute the second point to find the slope M

00:59 We'll isolate M

01:07 This is the function's slope

01:16 Now we'll substitute appropriate values to find the function equation

01:31 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given a function that is positive from the beginning of the axes. Plus the point (7,9) on the graph of the function. Find the equation for the function.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given information
Step 2: Calculate the slope of the line passing through the origin and the point (7, 9)
Step 3: Write the equation of the line that represents the function

Now, let's work through each step:
Step 1: We have the point (7, 9) and know the function is positive starting from the origin, suggesting a line through the origin.

Step 2: Calculate the slope $m$ :
Using the points (0, 0) and (7, 9), apply the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ :

$\quad m = \frac{9 - 0}{7 - 0} = \frac{9}{7}$

Step 3: Write the equation of the line:
Since the line passes through the origin, the form is $y = mx$ , so:

$\quad y = \frac{9}{7}x$

Therefore, the function equation is $y = 1\frac{2}{7}x$ .

Final Answer

$y=1\frac{2}{7}x$

Key Points to Remember

Essential concepts to master this topic

Origin Rule: Functions positive from origin pass through point (0,0)
Slope Formula: Use m = 9/7 from points (0,0) and (7,9)
Verification: Check that $\frac{9}{7} \times 7 = 9$ matches given point ✓

Common Mistakes

Avoid these frequent errors

Using the wrong points to calculate slope
Don't use random points or forget the origin = wrong slope calculation! Students often try to find slope without recognizing that "positive from the beginning" means the line starts at (0,0). Always identify that the function passes through the origin first, then use (0,0) and the given point.

Practice Quiz

Test your knowledge with interactive questions

Look at the function shown in the figure.

When is the function positive?

FAQ

Everything you need to know about this question

What does 'positive from the beginning of the axes' actually mean?

This means the function starts at the origin (0,0) and has positive y-values for positive x-values. It's telling you the line passes through the origin!

Why is the answer written as a mixed number instead of an improper fraction?

Both $\frac{9}{7}x$ and $1\frac{2}{7}x$ are correct! Mixed numbers are often preferred because they're easier to visualize - you can see it's slightly more than 1.

How do I convert between mixed numbers and improper fractions?

To convert $\frac{9}{7}$ to mixed: divide 9 ÷ 7 = 1 remainder 2, so $1\frac{2}{7}$ . To go back: 1 × 7 + 2 = 9, so $\frac{9}{7}$ .

Could this be a different type of function, not linear?

With only one point given plus the origin constraint, a linear function is the simplest answer. Other functions like $y = x^2$ or $y = \sqrt{x}$ are possible, but linear is most likely intended.

What if I calculated the slope as 7/9 instead of 9/7?

That's backwards! Remember: slope = $\frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$ . From (0,0) to (7,9): rise = 9, run = 7, so slope = $\frac{9}{7}$ .

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Find the Function Equation Passing Through Point (7,9): Positive Function Analysis

Linear Functions with Origin Constraints

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What does 'positive from the beginning of the axes' actually mean?

Why is the answer written as a mixed number instead of an improper fraction?

How do I convert between mixed numbers and improper fractions?

Could this be a different type of function, not linear?

What if I calculated the slope as 7/9 instead of 9/7?

🌟 Unlock Your Math Potential

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