These lessons introduce quadratic polynomials from a basic perspective. We then build on the notion of shifting basic parabolas into their vertex form. Completing the square is used as a fundamental tool in finding the turning point of a parabola. Finally, the zero product law is introduced as a way to find the zeroes of a quadratic function.
A parabola, as shown on the cables of the GoldenGate Bridge (below), can be seen in many different forms. The paththat a thrown ball takes or the flow of water from a hose each illustratethe shape of the parabola.
the shifted form of a parabola homework
The standard form is (x - h)2 = 4p (y - k), where the focusis (h, k + p) and the directrix is y = k - p. If the parabola is rotatedso that its vertex is (h,k) and its axis of symmetry is parallel to thex-axis, it has an equation of (y - k)2 = 4p (x - h), where thefocus is (h + p, k) and the directrix is x = h - p.
It would also be in our best interest to cover another form that theequation of a parabola may appear asy = (x - h)2 + k, where h represents the distance that the parabolahas been translated along the x axis, and k represents the distance theparabola has been shifted up and down the y-axis.
Let's first focus on the second form mentioned, y =(x - h)2+ k. When we have an equation in this form, we can safely say that the 'h'represents the same thing that 'h' represented in the first standard formthat we mentioned, as does the 'k'. When we have an equation like y = (x- 3)2 + 4, we see that the graph has been shifted 3 units tothe right and 4 units upward. The picture below shows this parabola in thefirst quadrant.
To find the line of symmetry of a parabola in this form, we need to rememberthat we are only dealing with parabolas that are pointed up or down in nature.With this in mind, the line of symmetry (also known as the axis of symmetry)is the line that splits the parabola into two separate branches that mirroreach other. The line of symmetry goes through the vertex, and since we arenow only dealing with parabolas that go up and down, the line of symmetrymust be a vertical line that will begin with "x = _ ". The numberthat goes in this blank will be the x-coordinate of the vertex. For example,when we looked at y = (x - 3)2 + 4, the x-coordinate of the vertexis going be 3; so the equation for the line of symmetry is x = 3.
As long as we have the equation in the form derived from the completingthe square step, we look and see if there is a negative sign in front ofthe parenthetical term. If the equation comes in the form of y = - (x -h)2 + k, the negative in front of the parenthesis tells us thatthe parabola is pointed downward (as illustrated in the picture below).If there is no negative sign in front, then the parabola faces upward.
From the above picture, I have labeled three items that we need to payclose attention to. The highest point of the parabola is the vertex (andthe maximum). The plus sign that is directly under the vertex is the focus.The green line that is above the parabola (and directly above the vertex)is the directrix. You may be able to see, by eyeballing, that the distancefrom the focus to the vertex is the same distance as the vertex to the directrix.We will now go into a bit of detail as to how to derive all of this informationfrom a given equation.
In order to find the focus and directrix of the parabola, we need tohave the equations that give an up or down facing parabola in the form (x- h)2 = 4p(y - k) form. In other words, we need to have the x2term isolated from the rest of the equation. We are used to having x2by itself, but if the vertex has been shifted either up or down, we needto show this in the parenthetical term with the y. The coefficient of the(y - k) term is the 4p term. We need to take this number and set it equalto 4p.
Now, we are going to begin taking what we have learned and start piecingit together. If we are given a focus and a vertex, we have enough to beable to generate a quadratic equation of a parabola. If we think about itfor a second, we will be able to find the distance from the vertex to thefocus based on this given information. We will then be able to calculateour p term (the term from the previous lesson that is in front of our non-squaredvariable). Placing the coordinates of the vertex into the equation is verysimple, relative to what we have learned so far.
The last pair of examples that we will examine will be one where we aregiven a quadratic equation that is not already in any particular standardform.We will now be forced to complete the square to arrive at the form we needto find the newest parts of the parabola that we have explored.
One reason we may want to identify the vertex of the parabola is that this point will inform us what the maximum or minimum value of the function is \((k)\), and where in the domain it occurs \((h)\).
The functions in parts (a) and (b) of Exercise 1 are examples of quadratic functions in standard form.When a quadratic function is in standard form, then it is easy to sketch its graph by reflecting, shifting, andstretching/shrinking the parabola y = x2.
We can translate the parabola vertically to produce a new parabola that is similar to the basic parabola. The function \(y=x^2+b\) has a graph which simply looks like the standard parabola with the vertex shifted \(b\) units along the \(y\)-axis. Thus the vertex is located at \((0,b)\). If \(b\) is positive, then the parabola moves upwards and, if \(b\) is negative, it moves downwards.
Similarly, we can translate the parabola horizontally. The function \(y=(x-a)^2\) has a graph which looks like the standard parabola with the vertex shifted \(a\) units along the \(x\)-axis. The vertex is then located at \((a,0)\). Notice that, if \(a\) is positive, we shift to the right and, if \(a\) is negative, we shift to the left.
In the module Algebra review , we revised the very important technique of completing the square. This method can now be applied to quadratics of the form \(y=x^2+qx+r\), which are congruent to the basic parabola, in order to find their vertex and sketch them quickly.
In fact, there is a similarity transformation that takes the graph of \(y=x^2\) to the graph of \(y=3x^2\). (Map the point \((x,y)\) to the point \((\dfrac13x, \dfrac13y)\).) Thus, the parabola \(y=3x^2\) is similar to the basic parabola.
This is an important point, since in the module Polynomials, it is seen that the graphs of higher degree equations (such as cubics and quartics) are not, in general, obtainable from the basic forms of these graphs by simple transformations. The parabola, and also the straight line, are special in this regard.
To graph parabolas with a vertex [latex]\left(h,k\right)[/latex] other than the origin, we use the standard form [latex]\left(y-k\right)^2=4p\left(x-h\right)[/latex] for parabolas that have an axis of symmetry parallel to the x-axis, and [latex]\left(x-h\right)^2=4p\left(y-k\right)[/latex] for parabolas that have an axis of symmetry parallel to the y-axis. These standard forms are given below, along with their general graphs and key features.
Start by writing the equation of the parabola in standard form. The standard form that applies to the given equation is [latex]\left(x-h\right)^2=4p\left(y-k\right)[/latex]. Thus, the axis of symmetry is parallel to the y-axis. To express the equation of the parabola in this form, we begin by isolating the terms that contain the variable [latex]x[/latex] in order to complete the square.
The vertex of the dish is the origin of the coordinate plane, so the parabola will take the standard form [latex]x^2=4py[/latex], where [latex]p>0[/latex]. The igniter, which is the focus, is 1.7 inches above the vertex of the dish. Thus we have [latex]p=1.7[/latex].
A parabola is a graph of a quadratic function. Pascal stated that a parabola is a projection of a circle. Galileo explained that projectiles falling under the effect of uniform gravity follow a path called a parabolic path. Many physical motions of bodies follow a curvilinear path which is in the shape of a parabola. In mathematics, any plane curve which is mirror-symmetrical and usually is of approximately U shape is called a parabola. Here we shall aim at understanding the derivation of the standard formula of a parabola, the different standard forms of a parabola, and the properties of a parabola.
There are four standard equations of a parabola. The four standard forms are based on the axis and the orientation of the parabola. The transverse axis and the conjugate axis of each of these parabolas are different. The below image presents the four standard equations and forms of the parabola.
The equation of the parabola can be derived from the basic definition of the parabola. A parabola is the locus of a point that is equidistant from a fixed point called the focus (F), and the fixed-line is called the Directrix (x + a = 0). Let us consider a point P(x, y) on the parabola, and using the formula PF = PM, we can find the equation of the parabola. Here the point 'M' is the foot of the perpendicular from the point P, on the directrix. Hence, the derived standard equation of the parabola is y2 = 4ax. 2ff7e9595c
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