Quadratic Function Practice Problems and Solutions

Understanding The Quadratic Function

Complete explanation with examples

$f\left(x\right)=ax^2+bx+c$

Where $a\ne0$ , since if the coefficient $a$ does not appear then it would not be a quadratic function.

The graph of a quadratic function will always be a parabola.

Example 1:

$f\left(x\right)=8x^2-2x+4$

It is a quadratic or second-degree function because its largest exponent is $2$ .

Example 2:

$f\left(x\right)=-7x^3+6x^2+2x-1$

This is not a second-degree function because, although it has an exponent $2$ , its largest exponent is $3$ .

Quadratic function

The equation of the basic quadratic function is: $y=ax^2+bx+c$

This way of writing them is called the general form of a second-degree function, where:

$ax^2$ is called the squared term, quadratic term, or second-degree term.

$a$ is the coefficient of the quadratic term.

$bx$ is called the linear term or first-degree term.

$b$ is the coefficient of the linear term.

$c$ is the constant term.

$x$ is called the unknown of the function and represents the number or numbers that make the function or in this case the equation true.

Example:

$f\left(x\right)=3x^2-5x+2$

$a=3$ , $b=-5$ and $c=2$

We must remember that for an equation to be of second degree, $a$ must always be different from zero.

Detailed explanation

Practice The Quadratic Function

Test your knowledge with 17 quizzes

$$ y=x^2 $$

Examples with solutions for The Quadratic Function

Step-by-step solutions included

Exercise #1

Determine the value of the coefficient $a$ in the following equation:

$-x^2+7x-9$

Step-by-Step Solution

The quadratic equation in the problem is already arranged (meaning all terms are on one side and 0 on the other side), so let's proceed to answer the question asked:

The question asked in the problem - What is the value of the coefficient $a$ in the equation?

Let's recall the definitions of coefficients in solving quadratic equations and the roots formula:

The rule states 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 $a$ is the coefficient of the quadratic term (meaning the term with the second power)- $x^2$ Let's examine the equation in the problem:

$-x^2+7x-9 =0$

Remember that the minus sign before the quadratic term means multiplication by: $-1$ , therefore- we can write the equation as:

$-1\cdot x^2+7x-9 =0$

The number that multiplies the $x^2$ , is $-1$ hence we identify that the coefficient of the quadratic term is the number $-1$ ,

Therefore the correct answer is A.

Answer:

-1

Video Solution

Exercise #2

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 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.

Answer:

8

Video Solution

Exercise #3

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

$4x^2+9x-2$

Step-by-Step Solution

The quadratic equation is given as $4x^2 + 9x - 2$ . This equation is in the standard form of a quadratic equation, which is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are coefficients.

The term $4x^2$ indicates that the coefficient $a = 4$ .
The term $9x$ indicates that the coefficient $b = 9$ .
The constant term $-2$ indicates that the coefficient $c = -2$ .

From this analysis, we can see that the coefficient $c$ is $-2$ .

Therefore, the value of the coefficient $c$ in the equation is $-2$ .

Answer:

-2

Video Solution

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 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.

Answer:

0

Video Solution

Exercise #5

$y=x^2+x+5$

Step-by-Step Solution

To solve this problem, we will identify the parameters of the given quadratic function step-by-step:

Step 1: Define the problem statement: We have $y = x^2 + x + 5$ .
Step 2: Identify the standard form of a quadratic function, which is $y = ax^2 + bx + c$ .
Step 3: Compare the given quadratic expression with the standard form to identify the coefficients.

Now, let's analyze the quadratic function provided:

From the given expression $y = x^2 + x + 5$ :
- The coefficient of $x^2$ is $1$ , so $a = 1$ .
- The coefficient of $x$ is $1$ , so $b = 1$ .
- The constant term is $5$ , so $c = 5$ .

Therefore, the parameters of the quadratic function are $a = 1$ , $b = 1$ , and $c = 5$ .

Consequently, the correct choice from the provided options is $(a = 1, b = 1, c = 5)$ .

Answer:

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

Video Solution

Frequently Asked Questions

Everything you need to know about The Quadratic Function

What is the standard form of a quadratic function?

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The standard form of a quadratic function is f(x) = ax² + bx + c, where a ≠ 0. The term ax² is the quadratic term, bx is the linear term, and c is the constant term.

How do you find the vertex of a quadratic function?

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You can find the vertex using the vertex form f(x) = a(x - h)² + k, where (h, k) is the vertex. Alternatively, use h = -b/(2a) to find the x-coordinate, then substitute to find the y-coordinate.

What's the difference between complete and incomplete quadratic functions?

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A complete quadratic function has all three terms: ax², bx, and c. An incomplete quadratic function is missing either the linear term (b = 0), constant term (c = 0), or both, but must always have the quadratic term ax².

How do you solve quadratic equations using the quadratic formula?

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Use the quadratic formula: x = (-b ± √(b² - 4ac))/(2a). First identify coefficients a, b, and c from ax² + bx + c = 0, then substitute into the formula to find the roots.

What does the discriminant tell you about quadratic solutions?

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The discriminant (b² - 4ac) determines the number of solutions: • If positive: two real solutions • If zero: one real solution • If negative: no real solutions (two complex solutions)

How do parabola transformations work in quadratic functions?

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Transformations follow these patterns: y = x² + k (vertical shift k units), y = (x - h)² (horizontal shift h units), and y = (x - h)² + k (combined horizontal and vertical shifts).

When should you use factoring vs quadratic formula?

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Use factoring when the quadratic easily factors into perfect squares or simple binomials. Use the quadratic formula for complex expressions, decimal coefficients, or when factoring isn't obvious.

How do you solve systems of quadratic equations?

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For two quadratic equations, use substitution or elimination methods. Graphically, solutions are intersection points of two parabolas. Systems can have 0, 1, or 2 solutions depending on how the parabolas intersect.

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Quadratic Function Practice Problems and Solutions

Understanding The Quadratic Function

Quadratic function

Practice The Quadratic Function

Examples with solutions for The Quadratic Function

Frequently Asked Questions

What is the standard form of a quadratic function?

How do you find the vertex of a quadratic function?

What's the difference between complete and incomplete quadratic functions?

How do you solve quadratic equations using the quadratic formula?

What does the discriminant tell you about quadratic solutions?

How do parabola transformations work in quadratic functions?

When should you use factoring vs quadratic formula?

How do you solve systems of quadratic equations?

More The Quadratic Function Questions

The function y=ax²+bx+c

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