Examples of horizontal translation in the following topics:
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- A translation of a function is a shift in one or more directions.
- In algebra, this essentially manifests as a vertical or horizontal shift of a function.
- To translate a function horizontally is the shift the function left or right.
- The general equation for a horizontal shift is given by:
- Let's use the same basic quadratic function to look at horizontal translations.
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- The image below shows an example of a function and its symmetry over the $x$-axis (vertical reflection) and over the $y$-axis (horizontal reflection).
- A function can have symmetry by reflecting its graph horizontally or vertically.
- This type of symmetry is a translation over an axis.
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- Recognize whether a function has an inverse by using the horizontal line test
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- There are three kinds of asymptotes: horizontal, vertical and oblique.
- Horizontal asymptotes of curves are horizontal lines that the graph of the function approaches as $x$ tends to $+ \infty$ or $- \infty$.
- Horizontal asymptotes are parallel to the $x$-axis.
- The $x$-axis is a horizontal asymptote of the curve.
- Hence, horizontal asymptote is given by:
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- If any horizontal line intersects the graph at more than one point, the function is not one-to-one.
- One way to check if the function is one-to-one is to graph the function and perform the horizontal line test.
- The graph below shows that it forms a parabola and fails the horizontal line test.
- Notice it fails the horizontal line test.
- Because the horizontal line crosses the graph of the function more than once, it fails the horizontal line test and cannot be one-to-one.
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- Now lets analyze horizontal scaling.
- This leads to a "shrunken" appearance in the horizontal direction.
- In general, the equation for horizontal scaling is:
- If $c$ is greater than one the function will undergo horizontal shrinking, and if $c$ is less than one the function will undergo horizontal stretching.
- If we want to induce horizontal shrinking, the new function becomes:
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- We will use the horizontal case to demonstrate how to determine the properties of an ellipse from its equation, so that $a$ is associated with x-coordinates, and $b$ with y-coordinates.
- For a horizontal ellipse, that axis is parallel to the $x$-axis.
- For a horizontal ellipse, it is parallel to the $y$-axis.
- For a horizontal ellipse, the foci have coordinates $(h \pm c,k)$, where the focal length $c$ is given by
- This diagram of a horizontal ellipse shows the ellipse itself in red, the center $C$ at the origin, the focal points at $\left(+f,0\right)$ and $\left(-f,0\right)$, the major axis vertices at $\left(+a,0\right)$ and $\left(-a,0\right)$, the minor axis vertices at $\left(0,+b\right)$ and $\left(0,-b\right)$.
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- Without any domain restriction, $f(x)=x^2$ does not have an inverse function as it fails the horizontal line test.
- But if we restrict the domain to be $x > 0$ then we find that it passes the horizontal line test and therefore has an inverse function.
- Notice that the parabola does not have a "true" inverse because the original function fails the horizontal line test and must have a restricted domain to have an inverse.
- This function fails the horizontal line test, and therefore does not have an inverse.
- However, if we restrict the domain to be $x>0$, then we find that it passes the horizontal line test and will match the inverse function.
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- The four main types of transformations are translations, reflections, rotations, and scaling.
- A translation moves every point by a fixed distance in the same direction.
- One possible translation of $f(x)$ would be $x^3 + 2$.
- This would then be read as, "the translation of $f(x)$ by two in the positive y direction".
- The function $f(x)=x^3$ is translated by two in the positive $y$ direction (up).
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- A horizontal reflection is a reflection across the $y$-axis, given by the equation: