Increasing Function Analysis: Connecting Two Fixed Points on a Coordinate Plane

Function Behavior with Point Coordinate Analysis

Is it possible to create an increasing function with the two given points?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Is it possible to create an increasing function from 2 points?
00:05 Let's look at 3 options
00:10 One option is constant
00:16 The second decreases then increases
00:22 And the third increases then decreases
00:31 Therefore it's impossible to draw an increasing function, and this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Is it possible to create an increasing function with the two given points?

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Step-by-step solution

To determine whether an increasing function can be created through the given two points, we must analyze and understand what conditions such a function satisfies.

An increasing function means that as xx increases, yy, the function’s value, also increases. In simple terms, if we have two points, (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), then for the function to be increasing, it must be that x1<x2x_1 < x_2 and y1<y2y_1 < y_2.

Starting from this understanding, observe the provided points. Assuming coordinates are (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) with specifics determined visually or contextually:

  • Verify that both points have distinct xx-coordinates such that x1<x2x_1 < x_2.
  • Check and confirm whether their respective yy-coordinates are such that y1<y2y_1 < y_2.

Upon observing plot arrangements, while the horizontal axis marks a left-right progression, the vertical arrangement negates: if corresponding plot layers detected inversely, no increase in height is shown relative to positional depth.

Therefore, observe if x1<x2x_1 < x_2 but y1>y2y_1 > y_2, causing the conclusion that such positional arrangement doesn't naturally derive an increasing function.

Conclusively, since this presented pattern arguably displays a decreasing nature, a true increasing function based on arrangement interpretation from two points is No.

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Final Answer

No

Key Points to Remember

Essential concepts to master this topic
  • Definition: Increasing function means if x1<x2x_1 < x_2, then y1<y2y_1 < y_2
  • Technique: Compare coordinates: left point (-4, 6), right point (6, 5)
  • Check: Since -4 < 6 but 6 > 5, function decreases ✓

Common Mistakes

Avoid these frequent errors
  • Assuming any line through two points is increasing
    Don't assume connecting two points always creates an increasing function = wrong conclusion! Students often think any function connecting points must go "up". Always check: does y increase when x increases?

Practice Quiz

Test your knowledge with interactive questions

Does the function in the graph decrease throughout?

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FAQ

Everything you need to know about this question

What exactly makes a function increasing?

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A function is increasing when moving from left to right, the function values (y-coordinates) get larger. If x1<x2x_1 < x_2, then we need y1<y2y_1 < y_2.

Can I just look at the slope to tell if it's increasing?

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Yes! If the slope between two points is positive, the function is increasing between those points. If the slope is negative, it's decreasing.

What if the points have the same y-coordinate?

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If both points have the same y-value, the function is constant (neither increasing nor decreasing) between those points. The slope would be zero.

Does this rule work for curved functions too?

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This specific rule applies to any function between two points! Whether the actual function is a line, curve, or step function, if it's increasing, the right point must be higher than the left point.

How do I read coordinates from a graph accurately?

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Look for grid lines and count carefully from the origin (0,0). The first number is the x-coordinate (horizontal), and the second is the y-coordinate (vertical).

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