Quadratic expressions and functions in general form ax2+bx+c, factored form a(x−α)(x−β),
The graph of a quadratic function has the general shape of a parabola.
It is symmetrical about the axis of symmetry and has a maxima or minima at the vertex, which lies on the axis of symmetry.
A quadratic in x is an expression of the form
where a=0.
The concavity of a parabola describes whether it "opens" up or down.
The parabola corresponding to ax2+bx+c is:
Concave up if a>0
Concave down if a<0.
Most quadratics can be factored as a product of linear terms:
We call the generalized form above factored form. Notice that α and β are the roots of the quadratic, since when x=α or x=β the expression will evaluate to zero.
We can factor quadratics in the form ax2+bx+c by splitting b into a sum α+β such that that αβ multiplies to ac.
After rewriting the expression as ax2+αx+βx+c, factor the first pair and the second pair separately. The common factor will emerge, and you can pull it out. The result is the fully-factored expression.
For example, in the quadratic 3x2+8x−3, we want to split 8 into α+β such that αβ=−9. We can do this by choosing α=9 and β=−1:
The roots of a quadratic correspond to the x-intercepts of its graph. When x=a or x=β, the entire expression equals zero, which is reflected on the graph.
The equation of the parabola below is −(x−α)(x−β):
Notice that the vertex still remains in the same place as we scale the parabola by a, but the squares are "stretched" depending on the value of a.
There are two things to notice:
The squares are being stretched into rectangles, but the number of shapes in each column is not changing, it's still (x−h)2. The height of each column is thus a(x−h)2.
The vertex does not move - it's height above the x-axis is still k.
All of this comes together to create our finished form:
A quadratic in the form ax2+bx+c can be written in the form
for some h,k∈R.
For the quadratic ax2+bx+c, the parabola has an axis of symmetry at
The axis of symmetry is the vertical line dividing the parabola perfectly in 2. The x-coordinate of the vertex, h, is equal to the x value where the axis of symmetry is located.
To convert from the form ax2+bx+c to a(x−h)2+k:
The values for a will match up directly.
Use the axis of symmetry x=−2ab to find h=−2ab.
Plug h into ax2+bx+c to find k.
Suppose we want to convert the parabola 3x2+12x−18 into vertex form.
First, identify the coefficients. We can see that a=3, b=12, and c=−18. The axis of symmetry is located at
In vertex form, the axis of symmetry is located at the x-coordinate of the vertex (h,k). So h=−2.
Since k is the y-coordinate of the vertex, we can plug x=−2 into 3x2+12x−18 to find it:
So in vertex form, the quadratic is
For any quadratic ax2+bx+c, the roots of the quadratic can be found using the quadratic formula:
When the values of a,b or c are large enough that using the quadratic formula becomes difficult, a calculator can be used to find the roots.
Your calculator should have an app for solving quadratics.
Nice work completing Quadratic Functions and parabolas, here's a quick recap of what we covered:
Exercises checked off
Quadratic expressions and functions in general form ax2+bx+c, factored form a(x−α)(x−β),
The graph of a quadratic function has the general shape of a parabola.
It is symmetrical about the axis of symmetry and has a maxima or minima at the vertex, which lies on the axis of symmetry.
A quadratic in x is an expression of the form
where a=0.
The concavity of a parabola describes whether it "opens" up or down.
The parabola corresponding to ax2+bx+c is:
Concave up if a>0
Concave down if a<0.
Most quadratics can be factored as a product of linear terms:
We call the generalized form above factored form. Notice that α and β are the roots of the quadratic, since when x=α or x=β the expression will evaluate to zero.
We can factor quadratics in the form ax2+bx+c by splitting b into a sum α+β such that that αβ multiplies to ac.
After rewriting the expression as ax2+αx+βx+c, factor the first pair and the second pair separately. The common factor will emerge, and you can pull it out. The result is the fully-factored expression.
For example, in the quadratic 3x2+8x−3, we want to split 8 into α+β such that αβ=−9. We can do this by choosing α=9 and β=−1:
The roots of a quadratic correspond to the x-intercepts of its graph. When x=a or x=β, the entire expression equals zero, which is reflected on the graph.
The equation of the parabola below is −(x−α)(x−β):
Notice that the vertex still remains in the same place as we scale the parabola by a, but the squares are "stretched" depending on the value of a.
There are two things to notice:
The squares are being stretched into rectangles, but the number of shapes in each column is not changing, it's still (x−h)2. The height of each column is thus a(x−h)2.
The vertex does not move - it's height above the x-axis is still k.
All of this comes together to create our finished form:
A quadratic in the form ax2+bx+c can be written in the form
for some h,k∈R.
For the quadratic ax2+bx+c, the parabola has an axis of symmetry at
The axis of symmetry is the vertical line dividing the parabola perfectly in 2. The x-coordinate of the vertex, h, is equal to the x value where the axis of symmetry is located.
To convert from the form ax2+bx+c to a(x−h)2+k:
The values for a will match up directly.
Use the axis of symmetry x=−2ab to find h=−2ab.
Plug h into ax2+bx+c to find k.
Suppose we want to convert the parabola 3x2+12x−18 into vertex form.
First, identify the coefficients. We can see that a=3, b=12, and c=−18. The axis of symmetry is located at
In vertex form, the axis of symmetry is located at the x-coordinate of the vertex (h,k). So h=−2.
Since k is the y-coordinate of the vertex, we can plug x=−2 into 3x2+12x−18 to find it:
So in vertex form, the quadratic is
For any quadratic ax2+bx+c, the roots of the quadratic can be found using the quadratic formula:
When the values of a,b or c are large enough that using the quadratic formula becomes difficult, a calculator can be used to find the roots.
Your calculator should have an app for solving quadratics.
Nice work completing Quadratic Functions and parabolas, here's a quick recap of what we covered:
Exercises checked off