How to Graph Parabolas in Shifted Form
A parabola is a curve that has a shape of a U or an inverted U. The standard form of a parabola is y = ax + bx + c, where a, b, and c are constants. The shifted form of a parabola is y = a(x - h) + k, where a, h, and k are constants. The shifted form is useful for graphing parabolas because it shows the vertex (the highest or lowest point) of the parabola as (h, k). The sign of a determines whether the parabola opens up (a > 0) or down (a ). The value of a also affects the width of the parabola: the larger the absolute value of a, the narrower the parabola.
theshiftedformofaparabolahomework
To graph a parabola in shifted form, follow these steps:
Identify the vertex (h, k) from the equation and plot it on the coordinate plane.
Determine whether the parabola opens up or down by looking at the sign of a.
Find the axis of symmetry, which is a vertical line that passes through the vertex. The equation of the axis of symmetry is x = h.
Choose two points on either side of the axis of symmetry and plug their x-coordinates into the equation to find their y-coordinates. Plot these points on the graph.
Draw a smooth curve that passes through the vertex and the other points.
Here is an example of graphing a parabola in shifted form:
Example: Graph y = -2(x + 3) + 4.
Solution:
The vertex is (-3, 4), so we plot this point on the graph.
The coefficient of x is -2, which means the parabola opens down.
The axis of symmetry is x = -3, so we draw a dashed line at this value.
We choose two points on either side of the axis of symmetry, such as (-4, 0) and (-2, 0). We plug these x-values into the equation to find their y-values: xy = -2(x + 3) + 4-4y = -2(-4 + 3) + 4y = -2(-1) + 4y = -2(1) + 4y = -2 + 4y = 2-2y = -2(-2 + 3) + 4y = -2(1) + 4y = -2(1) + 4y = -2 + 4y = 2We plot these points on the graph.
We draw a smooth curve that passes through the vertex and the other points.
The graph looks like this:
This article was
One of the applications of the shifted form of a parabola is to model real-world phenomena that have a maximum or minimum value, such as projectile motion, profit, height, etc. For example, if we want to find the maximum height of a ball thrown upward with an initial velocity of 20 m/s from a height of 1.5 m, we can use the equation y = -4.9(x - 2.04) + 21.6, where y is the height in meters and x is the time in seconds. The vertex of this parabola is (2.04, 21.6), which means that the ball reaches its maximum height of 21.6 m after 2.04 seconds.
You can also use the shifted form of a parabola to graph quadratic functions more easily by identifying the vertex and the axis of symmetry. For example, to graph y = 3(x + 1) - 5, you can plot the vertex (-1, -5) and draw the axis of symmetry x = -1. Then you can choose two points on either side of the axis of symmetry and plug their x-coordinates into the equation to find their y-coordinates. For example, if you choose x = -2 and x = 0, you get y = -2 and y = -2 respectively. Plot these points on the graph and draw a smooth curve that passes through them.
The shifted form of a parabola is also useful for finding the roots or zeros of a quadratic function, which are the x-values that make y equal to zero. To find the roots of y = a(x - h) + k, you can set y equal to zero and solve for x using the square root method. For example, to find the roots of y = -2(x + 3) + 4, you can do the following steps:
Set y equal to zero: -2(x + 3) + 4 = 0
Subtract 4 from both sides: -2(x + 3) = -4
Divide both sides by -2: (x + 3) = 2
Take the square root of both sides: x + 3 = Ââ2
Subtract 3 from both sides: x = -3 Ââ2
The roots are x = -3 +â2 and x = -3 -â2
The shifted form of a parabola is a convenient way to write and graph quadratic functions that have been translated horizontally or vertically. It also helps us find important features of a parabola such as the vertex, the axis of symmetry, and the roots. 0efd9a6b88
https://www.kawaiistaciemods.com/group/pol/discussion/c999efea-bab0-47aa-8897-a4c6f597f56b