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

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

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

Examples with solutions for The Quadratic Function

Step-by-step solutions included

Exercise #1

$y=x^2+10x$

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.

Answer:

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

Video Solution

Exercise #2

$y=2x^2+3$

Step-by-Step Solution

To solve this problem, we will follow these steps:

Step 1: Identify each term in the given function $y = 2x^2 + 3$ .
Step 2: Compare the equation to the standard quadratic form $y = ax^2 + bx + c$ .
Step 3: Determine the coefficients $a$ , $b$ , and $c$ .
Step 4: Match these coefficients to the correct multiple-choice option.

Step 1: The given function is $y = 2x^2 + 3$ . There is no $x$ term present.

Step 2: Compare this with the standard form $y = ax^2 + bx + c$ :

The coefficient of $x^2$ is $a = 2$ .
The coefficient of $x$ is $b = 0$ because there is no $x$ term.
The constant term is $c = 3$ .

Step 3: Therefore, the coefficients are $a = 2$ , $b = 0$ , and $c = 3$ .

Step 4: Review the multiple-choice options provided:

Choice 1: $a = 0$ , $b = 2$ , $c = 3$
Choice 2: $a = 0$ , $b = 3$ , $c = 2$
Choice 3: $a = 2$ , $b = 0$ , $c = 3$
Choice 4: $a = 3$ , $b = 0$ , $c = 2$

The correct choice is Choice 3: $a = 2$ , $b = 0$ , $c = 3$ .

Therefore, the solution to the problem is the values $a = 2$ , $b = 0$ , $c = 3$ which correspond to choice 3.

Answer:

$a=2,b=0,c=3$

Video Solution

Exercise #3

Identify the coefficients based on the following equation

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

Step-by-Step Solution

To solve this problem, we'll clearly delineate the given expression and compare it to the standard quadratic form:

Step 1: Recognize the standard form of a quadratic equation as $y = ax^2 + bx + c$ .
Step 2: Compare the given equation $y = x^2 - 6x + 4$ to the standard form.
Step 3: Identify coefficients:
- The coefficient of $x^2$ is $a = 1$ .
- The coefficient of $x$ is $b = -6$ .
- The constant term is $c = 4$ .

Therefore, the coefficients for the quadratic function $y = x^2 - 6x + 4$ are $a = 1$ , $b = -6$ , and $c = 4$ .

Among the provided choices, choice 3: $a=1,b=-6,c=4$ is the correct one.

Answer:

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

Video Solution

Exercise #4

Identify the coefficients based on the following equation

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

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Identify the given quadratic function.
Match it with the standard form of a quadratic equation $y = ax^2 + bx + c$ .
Extract the values of $a$ , $b$ , and $c$ directly from the comparison.

Now, let's work through each step:
Step 1: The given quadratic function is $y = 2x^2 - 3x - 6$ .
Step 2: The standard form of a quadratic equation is $y = ax^2 + bx + c$ .
Step 3: By matching the given quadratic function with the standard form:

- The coefficient of $x^2$ is $2$ , so $a = 2$ .
- The coefficient of $x$ is $-3$ , so $b = -3$ .
- The constant term is $-6$ , so $c = -6$ .

Therefore, the solution to the problem is $a = 2$ , $b = -3$ , $c = -6$ .

Answer:

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

Video Solution

Exercise #5

Identify the coefficients based on the following equation

$y=3x^2-81$

Step-by-Step Solution

To solve this problem, we will identify values of $a$ , $b$ , and $c$ in the quadratic function:

Step 1: Note the given equation $y = 3x^2 - 81$ .
Step 2: Compare it to the standard quadratic form $y = ax^2 + bx + c$ .
Step 3: Match coefficients to find $a$ , $b$ , and $c$ .

Now, let's work through each step:
Step 1: The given equation is $y = 3x^2 - 81$ .
Step 2: Compare this to the standard form, $y = ax^2 + bx + c$ . In this equation:
- The coefficient of $x^2$ is 3, hence $a = 3$ .
- There is no $x$ term, which means $b = 0$ .
- The constant term is $-81$ , hence $c = -81$ .

Therefore, the solution to the problem is $a = 3, b = 0, c = -81$ .

Answer:

$a=3,b=0,c=-81$

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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