Theshiftedformofaparabolahomework
Tanesha Prately
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 < 0). The value of a also affects the width of the parabola: the larger the absolute value of a, the narrower the parabola.
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:
x y = -2(x + 3) + 4 -4 y = -2(-4 + 3) + 4
y = -2(-1) + 4
y = -2(1) + 4
y = -2 + 4
y = 2-2 y = -2(-2 + 3) + 4
y = -2(1) + 4
y = -2(1) + 4
y = -2 + 4
y = 2
We 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:
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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.
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