What must apply for the vertex and direction of a parabola that exhibits the given characteristics?
B
b
Consider the equation y=(x-a)(x-a).
C
c
We can, for example, manipulate the answer from Part B.
A
a
Example Solution: y=- x^2-1
B
b
Example Solution: y=x^2-4x+4
C
c
Example Solution: y=x^2-4x-1
Practice makes perfect
a
If the quadratic equation has no x-intercept, it cannot intersect the x-axis. There are two ways that we can make this happen.
The curve has a vertexabove the x-axis. The curve must then open upwards not to intersect the x-axis. This means it has a positive x^2-term.
The curve has a vertex below the x-axis. The curve must then open downwards not to intersect the x-axis. This means it has a negative x^2-term.There is a second requirement. The curve must have a negative y-intercept. Since the negative part of the y-axis is below the x-axis, the vertex must be below the x-axis and the curve must open downwards. Let's start by showing the most basic of quadratic equations whose graph opens downwards, namely y=- x^2.
This quadratic has both an x- and a y-intercept at the origin. All we have to do now is to vertically translate the graph in the negative direction to make it not intersect the x-axis.
One example of a quadratic equation that fulfills the criteria is y=- x^2-1.
b
If a quadratic has one x-intercept, its vertex has to be on the x-axis. To give it a positive y-intercept its parabola also has to open upwards, meaning its x^2-term must have a positive coefficient. Any quadratic written in the following format will have a vertex that is on the x-axis at x=a and open upwards.
y=(x-a)(x-a)Let's arbitrarily choose a=2.
As we can see, our equation has one x-intercept and a positive y-intercept. If we multiply the parentheses, we can write the function in standard form.
One example of a quadratic equation that fulfills the criteria is y=x^2-4x+4.
c
In Part B, we created a quadratic with one x-intercept and a positive y-intercept. Let's have a look at it.
Notice that the graph intercepts the y-axis at y=4. To give this graph a negative y-intercept and two x-intercepts, we could vertically translate it downwards by more than 4 units. For example, if we subtract by 5 from the right-hand side the graph will shift down by 5 units, which gives it a y-intercept of y=- 1.
As we can see, our graph has two x-intercepts and a negative y-intercept. If we multiply the parentheses, we can write the function in standard form.
