# Parameter functions and graph plotting - Examples, Exercises and Solutions

## 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 –

1. Let's examine the parameter $a$ and ask: Is the function upward or downward facing?
2. Let's find the vertex of the function using the formula and then find the Y-coordinate of the vertex.
3. Let's find the points of intersection with the $X$ axis by substituting ($Y=0$).
4. 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.

## Practice Parameter functions and graph plotting

### Exercise #1

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

$3x^2+8x-5$

### Step-by-Step Solution

The quadratic equation of the problem is already arranged (that is, all the terms on one side and 0 on the other side), so we approach answering the question posed:

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

Let's remember the definitions of the coefficients when solving a quadratic equation and 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 examine the equation of the problem:

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

$x$ is

$8$And then we recognize b, which is the coefficient of the term in the first power, is the number$8$,

The correct answer is option d.

8

### Exercise #2

$y=x^2+10x$

### Step-by-Step Solution

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.

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

### Exercise #3

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

### Step-by-Step Solution

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.

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

### Exercise #4

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

$3x^2+5x$

### Step-by-Step Solution

The quadratic equation of the problem is already ordered (that is, all the terms on one side and 0 on the other side), so we approach answering the question posed:

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

Let's remember the definitions of the coefficients when solving a quadratic equation and 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 - that is, the coefficient of the term raised to the power of zero -$x^0$(And this is because any number other than zero raised to the power of zero equals 1:

$x^0=1$)

We examine the equation of the 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.

The correct answer is option c.

0

### Exercise #5

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

$-x^2+7x-9$

-1

### Exercise #1

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

$4x^2+9x-2$

-2

### Exercise #2

$y=x^2$

### Video Solution

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

### Exercise #3

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

### Video Solution

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

### Exercise #4

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

### Video Solution

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

### Exercise #5

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

### Video Solution

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

### Exercise #1

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

### Video Solution

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

### Exercise #2

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

$x^2=2x+7$

∓2

### Exercise #3

$y=x^2+x+5$

### Video Solution

$a=1,b=1,c=5$

### Exercise #4

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

### Video Solution

$a=-1,b=1,c=5$

### Exercise #5

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

### Video Solution

$a=3,b=-1,c=4$