the most basic quadratic function:
![Parabola y=X²](/_ipx/f_png,s_500x500/https://cdn.tutorela.com/images/1_Families_of_Parabolas.width-500.png)
the most basic quadratic function:
The family of parabolas
The basic quadratic function – with the addition of
In this family, we are given a quadratic function that clearly shows us how the function moves horizontally – how many steps it needs to move right or left.
represents the number of steps the function will move horizontally – right or left.
If is positive – (there is a minus in the equation) – the function will move steps to the right.
If is negative – (and as a result, there is a plus in the equation because minus times minus equals plus) – the function will move steps to the left.
In this quadratic function, we can see a combination of horizontal and vertical shifts:
: Determines the number of steps and the direction the function will move vertically – up or down.
positive – shift up, negative – shift down.
: Determines the number of steps and the direction the function will move horizontally – right or left.
Find the intersection of the function
\( y=(x+4)^2 \)
With the Y
Meet the most basic quadratic function:
In this function:
The function is a smiling minimum function and its vertex is -
The axis of symmetry of this function is .
The increasing interval of the function – all the values where the function is increasing are:
The decreasing interval of the function – all the values where the function is decreasing are:
Positive interval: all except – you can see in the graph that the entire function is above the axis
Negative interval: none. The entire function is above the axis.
Let's continue to a similar function from the same family:
In this function:
The function is a sad maximum function and its vertex is
The axis of symmetry of this function is .
Increasing interval:
Decreasing interval:
Positive interval: None. The entire function is below the axis.
Negative interval: All except
Let's continue with another function from the same family:
In this function:
Its vertex is-
The axis of symmetry of this function is .
The larger is: the parabola will have a smaller opening – closer to its axis of symmetry.
The smaller is: the parabola will have a larger opening – farther from its axis of symmetry.
The function from this family has no horizontal or vertical shift because in every function from this family the vertex is
Click here to practice and learn more about the function
The family of parabolas
The basic quadratic function – with the addition of
In this function:
– represents the intersection point of the function with the axis.
The meaning of is a vertical shift up or down of the function from the vertex .
If is positive – the function will shift vertically up by the number of steps indicated by .
If is negative – the function will shift vertically down by the number of steps indicated by .
This function describes only vertical shifts up and down according to
Let's see an example:
In this function -
this means that the intersection point of the function with the axis is .
And actually – the function moves vertically steps up.
Click here to practice and learn more about the function
In this family, we are given a quadratic function that clearly shows us how the function moves horizontally – how many steps it needs to move right or left.
represents the number of steps the function will move horizontally – right or left.
If is positive – (there is a minus in the equation) – the function will move steps to the right.
If is negative – (and as a result, there is a plus in the equation because minus times minus equals plus) – the function will move steps to the left.
Let's see an example:
This function moves from the vertex , steps to the right
Let's see this in the figure:
Click here to practice and learn more about the function
Find the intersection of the function
\( y=(x-2)^2 \)
With the X
What is the value of y for the function?
\( y=x^2 \)
of the point \( x=2 \)?
Complete:
The missing value of the function point:
\( f(x)=x^2 \)
\( f(?)=16 \)
In this quadratic function, we can see a combination of horizontal and vertical shifts:
: Determines the number of steps and the direction the function will move vertically – up or down.
positive – shift up, negative – shift down.
: Determines the number of steps and the direction the function will move horizontally – right or left.
Let's look at an example of combining two translations together:
For example, in the function:
The changes will be:
According to : the parabola will move steps to the left.
According to : the parabola will move steps up.
Let's see this in the illustration:
We can see that the vertex of the parabola is:
Click here to practice and learn more about the function
Find the ascending area of the function
\( f(x)=2x^2 \)
Find the descending area of the function
\( f(x)=\frac{1}{2}x^2 \)
Which chart represents the function \( y=x^2-9 \)?
What is the positive domain of the function below?
In the first step, we place 0 in place of Y:
0 = (x-2)²
We perform a square root:
0=x-2
x=2
And thus we reveal the point
(2, 0)
This is the vertex of the parabola.
Then we decompose the equation into standard form:
y=(x-2)²
y=x²-4x+2
Since the coefficient of x² is positive, we learn that the parabola is a minimum parabola (smiling).
If we plot the parabola, it seems that it is actually positive except for its vertex.
Therefore the domain of positivity is all X, except X≠2.
all x,
Find the intersection of the function
With the Y
Find the intersection of the function
With the X
What is the value of y for the function?
of the point ?
Complete:
The missing value of the function point: