Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [slope]

For questions on finding or applying slope, a number that describes both the direction and the steepness of a line.

In mathematics, the slope (or gradient) of a line is a number that describes both the direction and the steepness of the line. In the US and a few other countries the slop is denoted by the letter $m$ in the slope-intercept form of a line, $y=mx+b$.

Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.

429 questions
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Understanding the slope of a line as a rate of change

I thought I was confident about what is meant by the slope of a line and how it relates to a rate of change, but I'm having doubts and I'm hoping that someone will be able to help me clear them up. As I understand it, intuitively the slope of a line…
Feyn_example
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If $\sigma: \mathbb{R}\to \mathbb{R}^2$ is a function that spirals,goes to infinity and repeats itself, then is $\sigma$ non-injetive?

Let $\sigma:\mathbb{R}\to \mathbb{R}^2$ be a smooth function such that $$\frac{\text{d}\sigma}{\text{d}t}(s) \neq 0, \quad \forall s \in \mathbb{R},$$ and $$\sigma(t+n) = \sigma(n) +\sigma(t),\ \forall\ n \in \mathbb{Z}, \ t\in [0,1].…
Matheus Manzatto
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Quadratics: Intuitive relation between discriminant and derivative at roots

While working with quadratics that have real roots, I realized an interesting fact: The slope of a quadratic at its roots is equal to $\pm \sqrt{D}$ where $D=b^2-4ac$ Proof: $$f(x) = ax^2 + bx +c$$ $$f'(x) = 2ax+b$$ Roots: $$x =…
Vikhyat Agarwal
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How to find x and y coordinates based on the given distance?

The problem says: Find the point (coordinates $(x,y)=~?$) which is symmetrical to the point $(4,-2) $ considering the given equation $y=2x-3$ I have found the perpendicular line-slope $y=-~\frac{1}{2}x$ and the intersection point which is shown in…
Eugen Sunic
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Typo in book or I am wrong?

Find the equation of the tangent to the parabola $y = x^2$, if the $x$-intercept of the tangent is $2$. Now $$y = mx + b$$ $$0 = m(2) + 2$$ $$m = -1$$ so $$m = \frac{y-0}{x-2}$$ $$m(x-2) = y$$ $$-x-y = -2$$ but the book answer is $$8x-y =16$$
MrJonesBones
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True or False: The line y=x is always at 45 degrees to the x-axis?

I have a doubt. My teacher has posted a question that Is it always/sometimes/never true that the line y=x is at 45 degrees to the x-axis? I know that the slope of the line y=x is always 1, and the line must make 45 degrees with x-axis. I am a bit…
Jeet Kun
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If $|z|^2+\bar{A}z^2+A(\bar{z})^2+B\bar{z}+\bar{B}z+c=0$ represents a pair of intersecting lines.... find the value of $|A|$.

If $|z|^2+\bar{A}z^2+A(\bar{z})^2+B\bar{z}+\bar{B}z+c=0$ represents a pair of intersecting lines with angle of intersection $'\theta'$ then find the value of |A|. I tried using general equation of straight line in complex form as…
user69608
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Why derivative is a slope?

The change of $Y$ per $X$ is slope. And some say the change of slope per $X$ is derivative. So it is like slope of a slope! But slopes are always numbers like the slope of $2x$ is $2$. But derivates are not just numbers like the derivative of …
Anon Alexander
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Why limits give us the exact value of the slope of the tangent line?

Limits tell us how functions behave at $x\to a$, not how they behave at $x = a$. However, in limits we plug $x = a$ as an approximation of $x\to a$, so: why the limits give us the exact value of slope of the tangent line, despite being just an…
Mohammad Alshareef
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What is the equation for an ellipse given 3 points and the tangent line at those points?

Does anyone know how to find this? I know I have enough information to uniquely generate an ellipse (you only need five points for a conic, I have three plus the slopes or tangents at those points which count as points), but my problem is trying to…
Layke Findley
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Is the slope of a vertical line infinity or undefined?

By slope I mean derivative. Is something being infinity the same thing as being undefined? Is the slope of a vertical line negative infinity or positive infinity?
user477808
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Find slope using X-Intercept and Y-intercept

My mathematics teacher explained how to find slope using 2 different points but I was never taught how to use the X-Intercept and the Y-Intercept to find the slope. Example question: What is the slope of a line that has an X-Intercept of 8 and a…
Rawley Fowler
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Why $f(x) = x^2$ has variable derivative but its tangent has constant slope?

I'm taking Brilliant.org's calculus course, and I'm on the section called The Derivative. My (mis)understanding: A tangent line is a linear function that grazes a point, $a$, on the graph of a different function. The slope of that tangent line is…
user110391
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What is the tangent to this equation at origin?

$$x(x^2 + y^2) = a(x^2 - y^2)$$ I was trying to find the equation of the tangent as $$(Y-0) = \frac{dy}{dx}(X - 0)$$ where $$\frac{dy}{dx} = \frac{2ax-3x^2-y^2}{2(x+a)y}$$ So here putting the values of $(x,y)$ as $(0,0)$ makes the derivative non…
Ritoban Sen
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Find the slope of the line such that the area of triangle formed inside the circle is maximum

Here is the question. It states: A circle of radius 1 unit touches positive x -axis and positive y -axis at A and B respectively. A variable line passing through origin intersects the circle in two points D and E . If the area of the triangle…
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