Plotting the Quadratic Function Using Parameters a, b and c

Plotting the Quadratic Function

Plotting the graph of the quadratic function and examining the roles of the parameters $a, b, c$ in the function of the form $y = ax^2 + bx + c$

The quadratic function has three relevant characteristics:

$a$ – the coefficient of $X^2$ .
$b$ – the coefficient of $X$ .
$c$ – the constant term.

Steps to graph a quadratic function –

Let's examine the parameter $a$ and ask: Is the function upward or downward facing?
Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
Let's find the points of intersection with the $X$ axis by substituting ( $Y=0$ ).
Let's draw a coordinate system and first mark the vertex of the parabola.
Then, let's examine if the function is smiling or crying and mark the points of intersection with the $X$ axis that we found. Draw accordingly.

Detailed explanation

Start practice

Test yourself on plotting functions with parameters!

$y=x^2$

Incorrect

Correct Answer:

$a=1,b=0,c=0$

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Graphing the quadratic function ausing parameters a, b and c

Let's present the quadratic function and understand the meaning of each parameter in it.
$y=ax^2+bx+c$

$a$ – the coefficient of $X^2$
- -determines if the parabola will be a maximum or minimum parabola (sad or smiling) - determines the steepness – its width.

$a$ must be different from 0.
If $a$ is positive – the parabola is a minimum parabola – smiling
If $a$ is negative – the parabola is a maximum parabola – sad
The larger $a$ is – the narrower the function will be and vice versa.

$b$ - the coefficient of $X$
can be any number
indicates the position of the parabola along with $C$ .
determines the slope of the function at the intersection point with the $Y$ axis
(not relevant to the material)

$c$ – the constant term
is not dependent on $X$
can be any number
indicates the position of the parabola along with $X$
determines the y-intercept
responsible for the vertical shift of the function

Wonderful. Now, let's move on to plotting the quadratic function.

Steps to graph a quadratic function –

Let's examine the parameter $a$ and ask: is the function upward or downward facing?
Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
Let's find the points of intersection with the $X$ axis by substituting ( $Y=0$ ).
Let's draw a coordinate system and first mark the vertex of the parabola.
Then, let's examine if the function is smiling or sad and mark the points of intersection with the $X$ axis that we found. Draw accordingly.

Let's see an example -

In the following function:
$Y=-4X^2+4x+3$

$a=-4$ , negative. Therefore, the function is downward facing.
Let's find the vertex of the parabola:
$X=\frac{-4}{-4*2}=\frac{-4}{-8}=\frac{1}{2}$
$y=-4*\frac{1^2}{2}+4*\frac{1}{2}+3$
$y=4$
Vertex of the parabola $(\frac{1}{2}, 4)$
Let's find the intersection points with the $X$ axis
Substitute
$Y=0$
and we get:
$-4X^2+4x+3=0$
$X=-\frac{1}{2},1\frac{1}{2}$
Draw a coordinate system, mark relevant points, and draw logically: