Examples of point in the following topics:
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- The Cartesian coordinate system is used to visualize points on a graph by showing the points' distances from two axes.
- The point where the axes intersect is known as the origin.
- A Cartesian coordinate system is used to graph points.
- Points are specified uniquely in the Cartesian plane by a pair of numerical coordinates, which are the signed distances from the point to the two axes.
- Each point can be represented by an ordered pair $(x,y)
$, where the $x$-coordinate is the point's distance from the $y$-axis and the $y$-coordinate is the distance from the $x$-axis.
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- The point-slope form is ideal if you are given the slope and only one point, or if you are given two points and do not know what the $y$-intercept is.
- Given a slope, $m$, and a point $(x_{1}, y_{1})$, the point-slope equation is:
- Then plug this point into the point-slope equation and solve for $y$ to get:
- Example: Write the equation of a line in point-slope form, given point $(-3,6)$ and point $(1,2)$, and convert to slope-intercept form
- Plug this point and the calculated slope into the point-slope equation to get:
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- The distance and the midpoint formulas give us the tools to find important information about two points.
- The distance can be from two points on a line or from two points on a line segment.
- The distance between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is given by the formula:
- In geometry, the midpoint is the middle point of a line segment, or the middle point of two points on a line, and thus is equidistant from both end-points.
- If you have two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, the midpoint of the segment connecting the two points can be found with the formula:
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- The orange lines denote the distance between the focus and points on the conic section, as well as the distance between the same points and the directrix.
- From the definition of a parabola, the distance from any
point on the parabola to the focus is equal to the distance from that
same point to the directrix.
- In other words, the distance between a point on a conic section and its focus is less than the distance between that point and the nearest directrix.
- This indicates that the distance between a point on a conic section the nearest directrix is less than the distance between that point and the focus.
- Eccentricity is the ratio between the distance from any point on the
conic section to its focus, and the perpendicular distance from
that point to the nearest directrix.
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- Simply stated, this theorem points out that, if the plotted route between points A and C is smooth and continuous between point A to point C, you will have to pass through all points "B" on the journey, as long as they are on the plotted route.
- When you bike between points X and Z, and your path follows a semicircular route, you will bike through any point on a semicircle connecting points X and Z.
- This can be clearly seen by following the curve from point b.
- The function is defined for all real numbers x ≠ −2 and is continuous at every such point.
- In plotting a continuous and smooth function between two points, all points on the function between the extremes are described and predicted by the Intermediate Value Theorem.
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- We start by locating two points on the line.
- Locate two points on the graph, choosing points whose coordinates are integers.
- Starting with the point on the left, $(0, -3)$, sketch a right triangle, going from the first point to the second point, $(5, 1)$.
- Locate two points on the graph.
- Let $(x_1, y_1)$ be the point $(0, 5)$, and $(x_2, y_2)$ be the point $(3, 3)$.
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- The set of all points such that the difference between the distances to two focal points is constant
- The set of all points such that the ratio of the distance to a single focal point divided by the distance to a line (the directrix) is greater than one
- We want the set of all points that have the same difference between the distances to these points.
- Imagine that we take a point on the red hyperbola curve, called $P$, and we let that point be the $+a$ value on the $x$-axis.
- The ellipse can be defined as all points that have a constant sum of distances to two focal points, and the hyperbola is defined as all points that have constant difference of distances to two focal points.
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- Functions and relations can be symmetric about a point, a line, or an axis.
- The graph has symmetry over the origin or point $(0,0)$.
- The points given, $(1,3)$ and $(-1,-3)$ are reflected across the origin.
- The graph above has symmetry since the points labeled are reflected over the origin.
- Notice that the $x$-intercepts are reflected points over the axis of symmetry and are equidistant from the axis.
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- After creating a few $x$ and $y$ ordered pairs, we will plot them on the Cartesian plane and connect the points.
- We still don't have enough points to really see what's going on, so let's choose some more.
- So that's two new points: $(6,8)$ and $(6,-8)$.
- Connect these points with the best curve you can, and you'll discover you've drawn a parabola.
- The equation is the graph of a line through the three points found above.
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- In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
- Polar coordinates are points labeled $(r,θ)$ and plotted on a polar grid.
- This point is plotted on the grid in Figure.
- Points in the polar coordinate system with pole $0$ and polar axis $L$.
- In blue, the point $(4,210^{\circ})$.