Find the Linear Function Through (-3,-5) with 45-Degree Angle

Q: Why is the slope 1 when the angle is 45 degrees?

The slope equals tan(angle), and $ \tan(45°) = 1 $. This means for every 1 unit right, the line goes up 1 unit - a perfect diagonal!

Q: What if the angle was negative or greater than 90 degrees?

Negative angles give negative slopes (line goes down), and angles greater than 90° also give negative slopes. The tangent function handles all cases automatically.

Q: How do I remember the point-slope formula?

Think: "y minus y-point equals slope times (x minus x-point)" - $ y - y_1 = m(x - x_1) $. It shows how the line changes from the known point.

Q: What does it mean for a line to make an angle with the x-axis?

It's the angle of inclination - imagine standing at any point on the line and measuring the angle from the positive x-direction to the line itself, going counterclockwise.

Linear Functions with Angle Conditions

Which function represents a straight line that passes through the point $(-3,-5)$ and creates an angle of 45 degrees with the positive part of the x axis?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the algebraic representation of the function

00:03 We'll use the formula to calculate slope according to the angle with X-axis

00:07 We'll substitute the angle according to the given data and calculate to find the slope

00:17 This is the line's slope

00:22 Now we'll use the line equation

00:26 We'll substitute the point according to the given data

00:31 We'll substitute the slope and solve to find the intersection point (B)

00:49 We'll isolate the intersection point (B)

00:53 This is the intersection point with Y-axis

00:58 Now we'll substitute the intersection point and slope in the line equation

01:09 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

Which function represents a straight line that passes through the point $(-3,-5)$ and creates an angle of 45 degrees with the positive part of the x axis?

Step-by-step solution

To solve this problem, let's start by finding the slope $m$ of the line. Since the line makes a 45-degree angle with the positive $x$ -axis, the slope $m = \tan(45^\circ) = 1$ .

Next, we will use the point-slope form of the line equation, which is given by:

$y - y_1 = m(x - x_1)$

where $(x_1, y_1) = (-3, -5)$ and $m = 1$ . Substituting these values, we have:

$y + 5 = 1 \cdot (x + 3)$

which simplifies to:

$y + 5 = x + 3$

Subtract 5 from both sides to put it into slope-intercept form $y = mx + b$ , we get:

$y = x - 2$

Therefore, the function that represents the line through $(-3, -5)$ with a 45-degree angle with the $x$ -axis is $y = x - 2$ .

Final Answer

$y=x-2$

Key Points to Remember

Essential concepts to master this topic

Angle Rule: A 45° angle with x-axis gives slope m = tan(45°) = 1
Point-Slope Method: Use y - (-5) = 1(x - (-3)) to get y = x - 2
Verification: Check: point (-3, -5) satisfies y = x - 2 since -5 = -3 - 2 ✓

Common Mistakes

Avoid these frequent errors

Confusing angle with slope value
Don't use 45 as the slope directly = wrong equation y = 45x + b! The angle must be converted using tangent function first. Always use slope = tan(angle) to find the correct slope value.

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

Why is the slope 1 when the angle is 45 degrees?

The slope equals tan(angle), and $\tan(45°) = 1$ . This means for every 1 unit right, the line goes up 1 unit - a perfect diagonal!

What if the angle was negative or greater than 90 degrees?

Negative angles give negative slopes (line goes down), and angles greater than 90° also give negative slopes. The tangent function handles all cases automatically.

How do I remember the point-slope formula?

Think: "y minus y-point equals slope times (x minus x-point)" - $y - y_1 = m(x - x_1)$ . It shows how the line changes from the known point.

Can I use slope-intercept form directly instead?

You could, but you'd need to find the y-intercept first using $b = y_1 - mx_1$ . Point-slope form is usually faster when you have a specific point!

What does it mean for a line to make an angle with the x-axis?

It's the angle of inclination - imagine standing at any point on the line and measuring the angle from the positive x-direction to the line itself, going counterclockwise.

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