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Here are a few recommended readings before getting started with this lesson.
All three graphs of the functions presented earlier have a very distinctive form. These curves have a specific name, which now will be properly introduced.
Equations of parabolas always contain a variable raised to the second power. This is why the functions that represent vertical parabolas are called quadratic functions.
The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.
A parabola can intersect the x-axis at zero, one, or two points. Since the function's value at an x-intercept is always 0, these points are called zeros, or sometimes roots.
Because all graphs of quadratic functions extend infinitely to the left and right, they each have a y-intercept somewhere along the y-axis.
Consider the given parabola. Identify its zeros, line of symmetry, or y-intercept.
LaShay loves playing golf.
She is trying to improve her swing by drawing the parabola that the ball will make. Using her math knowledge, she has calculated the quadratic function that corresponds to this parabola.The axis of symmetry of the parabola is the vertical line through the vertex. Therefore, in this case, the equation of the axis of symmetry is x=23.
The y-intercept will now be determined and plotted. To do so, x=0 will be substituted in the given equation.x=0
Subtract term
(-a)2=a2
(ba)m=bmam
a⋅cb=ca⋅b
ba=b/2a/2
Add fractions
Mark is studying parabolas so that he can help LaShay with her golf swing. He wants to write the vertex form of the quadratic function that corresponds to the given graph.
Help Mark find the desired equation!Start by identifying the vertex of the parabola.
x=0, y=1
Subtract term
(-a)2=a2
LHS+2=RHS+2
LHS/9=RHS/9
ba=b/3a/3
Rearrange equation
In the following applet, several quadratic functions are expressed in different forms. Are the quadratic functions written in vertex form, factored form, or neither?
By now, it is not a secret that LaShay loves playing golf.
She is once again trying to improve her swing by drawing the parabola that the ball will make. One more time, she uses her math knowledge to calculate the quadratic function that corresponds to this parabola.Note that, for the given function, the value of a is -2. Therefore, the parabola opens downward. This corresponds to the loci of the points plotted in the coordinate plane. Finally, these points can be connected with a smooth curve to draw the parabola.
Continuing his studies, Mark wants to write the factored form of the quadratic function that corresponds to the given parabola.
Help Mark find the desired equation!Start by identifying the x-intercepts of the parabola.
x=-2, y=-3
Add and subtract terms
a(-b)=-a⋅b
LHS/(-9)=RHS/(-9)
-b-a=ba
ba=b/3a/3
Rearrange equation
For the following quadratic functions, identify the vertex or the zeros.
(a−b)2=a2−2ab+b2
\CommutativePropMult
Multiply
Calculate power
Distribute -2
Distribute (x−3)
Distribute -2x & 2
Add terms