X-Axis Intercepts: Find Ordered Pairs For F(x) = (1/2)x^2 + X - 9

by SLV Team 66 views
X-Axis Intercepts: Find Ordered Pairs for f(x) = (1/2)x^2 + x - 9

Hey guys! Today, we're diving into a fun math problem where we need to figure out where the graph of a quadratic function crosses the negative x-axis. Specifically, we're working with the function f(x) = (1/2)x^2 + x - 9. This might sound a bit intimidating at first, but trust me, we'll break it down step by step so it's super easy to understand. Our main goal is to pinpoint the two ordered pairs between which the graph dips below the x-axis on the negative side. So, grab your thinking caps, and let's get started!

Understanding the Problem

First, let's make sure we all understand what the question is asking. We're given a quadratic function, which means its graph will be a parabola – a U-shaped curve. The x-axis is the horizontal line where y = 0. The points where the parabola crosses the x-axis are called the x-intercepts, also known as the roots or zeros of the function. We're specifically interested in where the graph crosses the negative x-axis, meaning the x-values will be negative numbers. We need to find two ordered pairs (x, 0) – since the y-coordinate will be 0 at the x-axis – between which the graph transitions from being above the x-axis to below it, or vice versa, on the negative side of the x-axis.

To solve this, we're essentially looking for two consecutive x-values where the function changes sign. If f(x) is positive for one x-value and negative for the next, then the graph must cross the x-axis somewhere in between those two x-values. We can test the given ordered pairs in the options to see which pair brackets a sign change. This involves plugging in the x-values from the options into our function, f(x) = (1/2)x^2 + x - 9, and observing the resulting y-values. If we find a pair where one y-value is positive and the other is negative, we've found our answer. This method is a straightforward way to tackle this problem without needing to solve the quadratic equation directly.

Step-by-Step Solution

Okay, let's dive into the actual solution. We'll take each option and plug in the x-values into our function, f(x) = (1/2)x^2 + x - 9, to see what y-values we get. Remember, we're looking for a sign change, meaning one positive and one negative y-value within the pair.

Option A: (-6, 0) and (-5, 0)

Let's start with x = -6:

f(-6) = (1/2)(-6)^2 + (-6) - 9 f(-6) = (1/2)(36) - 6 - 9 f(-6) = 18 - 6 - 9 f(-6) = 3

Now, let's try x = -5:

f(-5) = (1/2)(-5)^2 + (-5) - 9 f(-5) = (1/2)(25) - 5 - 9 f(-5) = 12.5 - 5 - 9 f(-5) = -1.5

Hey, look at that! We've got a sign change. f(-6) is 3 (positive), and f(-5) is -1.5 (negative). This means the graph crosses the x-axis somewhere between x = -6 and x = -5. So, it looks like option A is our winner!

Checking Other Options (Just to be sure)

Even though we've found a potential answer, it's always good to double-check the other options to make sure we haven't missed anything.

Option B: (-3, 0) and (-2, 0)

f(-3) = (1/2)(-3)^2 + (-3) - 9 f(-3) = (1/2)(9) - 3 - 9 f(-3) = 4.5 - 3 - 9 f(-3) = -7.5

f(-2) = (1/2)(-2)^2 + (-2) - 9 f(-2) = (1/2)(4) - 2 - 9 f(-2) = 2 - 2 - 9 f(-2) = -9

Both f(-3) and f(-2) are negative, so no sign change here.

Option C: (-2, 0) and (-1, 0)

We already know f(-2) = -9 from the previous step. Let's calculate f(-1):

f(-1) = (1/2)(-1)^2 + (-1) - 9 f(-1) = (1/2)(1) - 1 - 9 f(-1) = 0.5 - 1 - 9 f(-1) = -9.5

Again, both values are negative, so no sign change.

Option D: (-4, 0) and (-3, 0)

f(-4) = (1/2)(-4)^2 + (-4) - 9 f(-4) = (1/2)(16) - 4 - 9 f(-4) = 8 - 4 - 9 f(-4) = -5

We already know f(-3) = -7.5 from option B. Both are negative, so no sign change.

The Final Answer

Alright, guys, after carefully checking each option, it's clear that the graph of f(x) = (1/2)x^2 + x - 9 crosses the negative x-axis between the ordered pairs (-6, 0) and (-5, 0). Option A is definitely the correct answer!

Key Takeaways

So, what did we learn from this problem? Here are a few key takeaways:

  • Understanding x-intercepts: The x-intercepts are the points where the graph of a function crosses the x-axis (where y = 0). They are also known as the roots or zeros of the function.
  • Sign Changes and X-Axis Crossing: A sign change in the function's value (from positive to negative or vice versa) between two x-values indicates that the graph crosses the x-axis somewhere between those values.
  • Testing Ordered Pairs: When faced with multiple-choice questions like this, plugging in the given values and checking for sign changes can be a highly effective strategy.
  • Quadratic Functions and Parabolas: Quadratic functions create parabolas, U-shaped curves that can cross the x-axis at zero, one, or two points.

Tips for Similar Problems

If you encounter a similar problem in the future, here are some tips to keep in mind:

  • Visualize the Graph: Try to picture the graph of the function in your mind. This can help you understand what the question is asking and what a reasonable answer might look like.
  • Use the Sign Change Method: If you're looking for intervals where a function crosses the x-axis, checking for sign changes is a reliable technique.
  • Don't Be Afraid to Plug In Values: Plugging in values can often be a straightforward way to narrow down your options, especially in multiple-choice questions.
  • Double-Check Your Work: Always take a moment to double-check your calculations to avoid making small errors.
  • Consider Graphing Tools: If you have access to a graphing calculator or online graphing tool, use it to visualize the function and confirm your answer.

Wrapping Up

Great job, guys! We've successfully navigated through this problem and found the ordered pairs where the graph of our function crosses the negative x-axis. Remember, the key is to break down the problem into manageable steps, understand the concepts involved, and use the given options to your advantage. Keep practicing, and you'll become a pro at these types of problems in no time! Now, go forth and conquer more math challenges!

In conclusion, to find the ordered pairs where the graph of f(x) = (1/2)x^2 + x - 9 crosses the negative x-axis, we evaluated the function at the x-values provided in the options. By identifying the interval where the function's value changes sign, we determined that the graph crosses the x-axis between (-6, 0) and (-5, 0). This method of checking for sign changes is a valuable tool for analyzing functions and understanding their behavior. The importance of understanding x-intercepts cannot be overstated, as they provide critical information about the function's roots or zeros. Remember, the x-intercepts are the points where the function's graph intersects the x-axis, and finding these points helps us grasp the overall shape and position of the graph. Specifically, evaluating f(x) at x = -6 and x = -5 gave us function values of 3 and -1.5, respectively, confirming a sign change and thus, the x-axis crossing. This methodical approach, combined with a solid understanding of quadratic functions, makes these types of problems much more approachable. For future problems, always consider visualizing the graph, applying the sign change method, and verifying your calculations to ensure accuracy. The concept of a parabola is central to understanding the graph of a quadratic function, and knowing that parabolas can cross the x-axis at zero, one, or two points is key to solving problems like this one. Don't forget to utilize all the tools at your disposal, including graphing calculators or online tools, to visually confirm your answers and deepen your understanding.