y-axis
The axis on a graph that is usually drawn from bottom to top, with values increasing farther up.
Examples of y-axis in the following topics:
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Reflections
- Reflections are a type of transformation that move an entire curve such that its mirror image lies on the other side of the or $y$-axis.
- The reflection of a function can be performed along the -axis, the $y$-axis, or any line.
- A horizontal reflection is a reflection across the $y$-axis, given by the equation:
- The result is that the curve becomes flipped over the $y$-axis.
- Calculate the reflection of a function over the -axis, $y$-axis, or the line $y=x$
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The Cartesian System
- The horizontal axis is known as the -axis, and the vertical axis is known as the $y$-axis.
- Each point can be represented by an ordered pair $(x,y) xy$-axis and the $yx$-axis.
- On the -axis, numbers increase toward the right and decrease toward the left; on the $y$-axis, numbers increase going upward and decrease going downward.
- The non-integer coordinates lie between -1 and -2 on the -axis and between -2 and -3 on the $y$-axis.
- The revenue is plotted on the $y$-axis, and the number of cars washed is plotted on the -axis.
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Basics of Graphing Exponential Functions
- As you connect the points, you will notice a smooth curve that crosses the $y$-axis at the point and is increasing as takes on larger and larger values.
- As you can see in the graph below, the graph of $y=\frac{1}{2}^xy=2^xy$-axis.
- That is, if the plane were folded over the $y$-axis, the two curves would lie on each other.
- The function $y=b^xx$-axis as a horizontal asymptote because the curve will always approach the -axis as approaches either positive or negative infinity, but will never cross the axis as it will never be equal to zero.
- The graph of this function crosses the $y$-axis at and increases as approaches infinity.
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Graphing Equations
- For an equation with two variables, and $yx$-axis and a $y$-axis.
- We will use the Cartesian plane, in which the -axis is a horizontal line and the $y$-axis is a vertical line.
- For the three values for , let's choose a negative number, zero, and a positive number so we include points on both sides of the $y$-axis:
- $\begin{aligned} (0)^{2}+y^{2} &= 100 \\ y^{2} &= 100 \\ \sqrt{y^{2}}&=\sqrt{100} \\ y &= \pm10 \end{aligned}$
- $\begin{aligned} (6)^2+y^2&=100 \\ 36+y^2&=100 \\ 36+y^2-36&=100-36 \\ y^2&=64 \\ y&=\pm 8 \end{aligned}$
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Double Integrals Over General Regions
- the projection of onto either the -axis or the $y$-axis is bounded by the two values, and .
- -axis: If the domain is normal with respect to the -axis, and is a continuous function, then and (defined on the interval ) are the two functions that determine .
- $y$-axis: If is normal with respect to the $y$-axis and is a continuous function, then $\alpha(y)$ and $\beta(y)$ (defined on the interval $[a, b]D$.
- $\displaystyle{\iint_D f(x,y)\ dx\, dy = \int_a^b dy \int_{\alpha (y)}^{ \beta (y)} f(x,y)\, dx}$
- $D = \{ (x,y) \in \mathbf{R}^2 \ : \ x \ge 0, y \le 1, y \ge x^2 \}$
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Zeroes of Linear Functions
- Graphically, where the line crosses the -axis, is called a zero, or root.
- If there is a horizontal line through any point on the $y$-axis, other than at zero, there are no zeros, since the line will never cross the -axis.
- If the horizontal line overlaps the -axis, (goes through the $y$-axis at zero) then there are infinitely many zeros, since the line intersects the -axis multiple times.
- To find the zero of a linear function algebraically, set $y=0$ and solve for $x$.
- The blue line, $y=\frac{1}{2}x+2(-4,0)y=-x+5(5,0)$.
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Graphing Quadratic Equations In Standard Form
- The axis of symmetry for a parabola is given by:
- Because and the axis of symmetry is:
- More specifically, it is the point where the parabola intercepts the y-axis.
- Note that the parabola above has and it intercepts the $y$-axis at the point
- The axis of symmetry is a vertical line parallel to the y-axis at .
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Area of a Surface of Revolution
- If the curve is described by the function $y = f(x) (a≤x≤b)A_yA_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dxx$-axis.
- Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is co-planar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis.
- If the curve is described by the parametric functions , $y(t)t$ ranging over some interval $[a,b]$ and the axis of revolution the $y$-axis, then the area $A_y$ is given by the integral:
- Likewise, when the axis of rotation is the -axis, and provided that $y(t)$ is never negative, the area is given by:
- A portion of the curve rotated around the -axis (vertical in the figure).
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Introduction to Ellipses
- To do this, we introduce a scaling factor into one or both of the -$y$ coordinates.
- This has the effect of stretching the ellipse further out on the -axis, because larger values of are now the solutions.
- Now all the $y$ values are stretched vertically, further away from the origin.
- The ellipse $x^2 +\left( \frac{y}{3} \right)^2 = 1$ has been stretched along the $y$-axis by a factor of 3 as compared to the circle $x^2 + y^2 = 1$.
- The ellipse $\left( \frac{x}{3} \right)^2 +y^2 = 1$ has been stretched along the $x$-axis by a factor of 3 as compared to the circle $x^2 + y^2 = 1$.
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Cylindrical Shells
- In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.
- The volume of the solid formed by rotating the area between the curves of and and the lines and about the $y$-axis is given by:
- The volume of solid formed by rotating the area between the curves of $f(y)y=ay=bx$-axis is given by:
- $\displaystyle{V = 2\pi \int_a^b x \left | f(y) - g(y) \right | \,dy}$
- Each segment located at , between and the -axis, gives a cylindrical shell after revolution around the vertical axis.