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

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

$a$ – the coefficient of $X^2$.

$b$ – the coefficient of $X$.

$c$ – the constant term.

- 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.

Question 1

\( y=2x^2-5x+6 \)

Question 2

\( y=x^2+10x \)

Question 3

What is the value of the coefficient \( b \) in the equation below?

\( 3x^2+8x-5 \)

Question 4

What is the value of the coefficient \( c \) in the equation below?

\( 3x^2+5x \)

Question 5

\( y=-2x^2+3x+10 \)

$y=2x^2-5x+6$

In fact, a quadratic equation is composed as follows:

y = ax²-bx-c

That is,

a is the coefficient of x², in this case 2.

b is the coefficient of x, in this case 5.

And c is the number without a variable at the end, in this case 6.

$a=2,b=-5,c=6$

$y=x^2+10x$

Here we have a quadratic equation.

A quadratic equation is always constructed like this:

$y = ax²+bx+c$

Where a, b, and c are generally already known to us, and the X and Y points need to be discovered.

Firstly, it seems that in this formula we do not have the C,

Therefore, we understand it is equal to 0.

$c = 0$

a is the coefficient of X², here it does not have a coefficient, therefore

$a = 1$

$b= 10$

is the number that comes before the X that is not squared.

$a=1,b=10,c=0$

What is the value of the coefficient $b$ in the equation below?

$3x^2+8x-5$

The quadratic equation of the given problem is already arranged (that is, all the terms are found on one side and the 0 on the other side), thus we approach the given problem as follows;

**In the problem, the question was asked: **what is the value of the coefficient$b$in the equation?

**Let's remember** the definitions of coefficients when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$__are :__

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

**That is **the coefficient$b$is the coefficient of the **term in the first power** -$x$**We then examine **the equation of the given problem:

$3x^2+8x-5 =0$That is, the number that multiplies

$x$ is

$8$Consequently we are able to identify b, which is the coefficient of the term in the first power, as the number$8$,

__Thus the correct answer is option d.__

8

What is the value of the coefficient $c$ in the equation below?

$3x^2+5x$

The quadratic equation of the given problem has already been arranged (that is, all the terms are on one side and 0 is on the other side) thus we can approach the question as follows:

**In the problem, the question was asked: **what is the value of the coefficient$c$in the equation?

**Let's remember** the definition of a coefficient when solving a quadratic equation as well as the formula for the roots:

The rule says that the roots of an equation of the form

$ax^2+bx+c=0$__are:__

$x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

**That is **the coefficient

$c$is the free term - and as such the coefficient of the **term is raised to the power of zero** -$x^0$(Any number other than zero raised to the power of zero equals 1:

$x^0=1$)

**Next we examine **the equation of the given problem:

$3x^2+5x=0$Note that there is no free term in the equation, that is, the numerical value of the free term is 0, in fact the equation can be written as follows:

$3x^2+5x+0=0$and therefore the value of the coefficient$c$ is 0.

__Hence the correct answer is option c.__

0

$y=-2x^2+3x+10$

$a=-2,b=3,c=10$

Question 1

\( y=2x^2-3x-6 \)

Question 2

\( y=3x^2+4x+5 \)

Question 3

\( y=x^2-6x+4 \)

Question 4

\( y=x^2 \)

Question 5

What is the value ofl coeficiente \( a \) in the equation?

\( -x^2+7x-9 \)

$y=2x^2-3x-6$

$a=2,b=-3,c=-6$

$y=3x^2+4x+5$

$a=3,b=4,c=5$

$y=x^2-6x+4$

$a=1,b=-6,c=4$

$y=x^2$

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

What is the value ofl coeficiente $a$ in the equation?

$-x^2+7x-9$

-1

Question 1

What is the value of the coefficient \( c \) in the equation below?

\( 4x^2+9x-2 \)

Question 2

Choose the correct algebraic expression based on the parameters:

\( a=-3,b=3,c=7 \)

Question 3

Create an algebraic expression based on the following parameters:

\( a=-1,b=-1,c=-1 \)

Question 4

Create an algebraic expression based on the following parameters:

\( a=0,b=1,c=0 \)

Question 5

Create an algebraic expression based on the following parameters:

\( a=-1,b=0,c=0 \)

What is the value of the coefficient $c$ in the equation below?

$4x^2+9x-2$

-2

Choose the correct algebraic expression based on the parameters:

$a=-3,b=3,c=7$

$-3x^2+3x+7$

Create an algebraic expression based on the following parameters:

$a=-1,b=-1,c=-1$

$-x^2-x-1$

Create an algebraic expression based on the following parameters:

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

$x$

Create an algebraic expression based on the following parameters:

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

$-x^2$