Identify Quadratic Function Leading Coefficient 3 Constant Term -12

Hey guys! Let's dive into the world of quadratic functions and figure out which one fits the bill with a leading coefficient of 3 and a constant term of -12. Quadratic functions are super important in math, and understanding their parts can really help you solve problems. We're going to break down what a leading coefficient and a constant term are, and then we'll go through each option to find the correct one. So, buckle up and let's get started!

Understanding Quadratic Functions

First off, let's talk about what a quadratic function actually is. A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is 2. The general form of a quadratic function is:

f(x) = ax^2 + bx + c

Where:

• f(x) represents the function's value at x
• a, b, and c are constants, with a not equal to 0 (otherwise, it wouldn't be quadratic!)
• x is the variable

Now, let’s break down the key parts that we need to focus on for this problem: the leading coefficient and the constant term.

Leading Coefficient

The leading coefficient is the number that's multiplied by the x^2 term. In the general form above, the leading coefficient is a. This number is super important because it tells us a lot about the parabola (the U-shaped graph of a quadratic function). For example, if a is positive, the parabola opens upwards, and if a is negative, it opens downwards. In our case, we're looking for a quadratic function with a leading coefficient of 3, meaning a should be 3. Keep an eye out for this when we go through the options!

Constant Term

The constant term is the number that stands alone without any x attached. In the general form f(x) = ax^2 + bx + c, the constant term is c. This term tells us where the parabola intersects the y-axis (also known as the y-intercept). For this problem, we need a constant term of -12, so we're looking for c = -12. Got it? Great!

Analyzing the Options

Okay, now that we know what we're looking for, let's go through the options and see which one matches our criteria. We need a quadratic function where the leading coefficient is 3 and the constant term is -12. Let's break down each option:

Option A: $f(x)=-12 x^2+3 x+1$

In option A, the function is $f(x) = -12x^2 + 3x + 1$. Let's identify the leading coefficient and the constant term:

• Leading coefficient: The number multiplying the $x^2$ term is -12.
• Constant term: The number without any $x$ is 1.

Does this match our requirements? We need a leading coefficient of 3, but this one has -12. We also need a constant term of -12, but this one has 1. So, option A is not the correct answer. Nice try, option A, but you didn't quite make the cut!

Option B: $f(x)=3 x^2+11 x-12$

Let's take a look at option B, which is $f(x) = 3x^2 + 11x - 12$. Again, we'll find the leading coefficient and the constant term:

• Leading coefficient: The number multiplying the $x^2$ term is 3.
• Constant term: The number without any $x$ is -12.

Hold on a second! Does this look familiar? The leading coefficient is indeed 3, and the constant term is -12. Bingo! Option B seems to fit our criteria perfectly. But, just to be sure, let's check the other options too. We want to be absolutely certain we've got the right answer.

Option C: $f(x)=13 x^2+3 x+3$

Moving on to option C, we have $f(x) = 13x^2 + 3x + 3$. Let's identify those key terms:

• Leading coefficient: The number multiplying the $x^2$ term is 13.
• Constant term: The number without any $x$ is 3.

This one doesn't match our requirements either. We need a leading coefficient of 3, but this has 13. And we need a constant term of -12, but this one has 3. Option C, you're out!

Option D: $f(x)=3 x-12$

Finally, let’s analyze option D, $f(x) = 3x - 12$. At first glance, something might seem a bit off here. Remember, we're looking for a quadratic function, which means it needs an $x^2$ term. Let's break it down anyway:

• Leading coefficient: Wait a minute… there’s no $x^2$ term! This is actually a linear function, not a quadratic one.
• Constant term: The number without any $x$ is -12.

Option D has a constant term of -12, which is correct, but it's missing the $x^2$ term, so it can't be our answer. Sorry, option D, you're close but no cigar!

Conclusion

Alright, guys, we’ve gone through all the options and broken them down piece by piece. We were looking for a quadratic function with a leading coefficient of 3 and a constant term of -12. After analyzing each option:

• Option A had a leading coefficient of -12 and a constant term of 1.
• Option B had a leading coefficient of 3 and a constant term of -12.
• Option C had a leading coefficient of 13 and a constant term of 3.
• Option D was a linear function, not a quadratic one.

So, the winner is Option B: $f(x) = 3x^2 + 11x - 12$.

Option B perfectly fits the criteria: a leading coefficient of 3 and a constant term of -12. Great job, everyone! Understanding the parts of a quadratic function—like the leading coefficient and the constant term—is super useful for solving problems like this. Keep practicing, and you'll become quadratic function pros in no time!

The correct answer is B. $f(x)=3 x^2+11 x-12$