Исследование функции. Часть 3. Точки пересечения с осями координат

Main ideas / lessons

• The lesson is part of “studying functions,” specifically the third point: finding where the graph of a function intersects the coordinate axes.
• Key definitions:
  • x-axis = horizontal axis.
  • y-axis = vertical axis.
• Core method:
  • To find x-intercepts (intersection with the x-axis), set the function value equal to 0.
  • To find y-intercepts (intersection with the y-axis), set x = 0.

Methodology / step-by-step instructions

Goal 1: Intersection with the x-axis

1. Start with the function: [ y = \frac{x^2 - 4}{x+3} ]

2. Set (y = 0): [ 0 = \frac{x^2 - 4}{x+3} ]

3. For a rational equation:

   • The numerator may be 0,
   • The denominator must not be 0 (division by zero is forbidden).

4. Therefore: [ x^2 - 4 = 0 ] and require: [ x+3 \ne 0 \quad (\text{i.e., } x \ne -3) ]

5. Solve the quadratic: [ x^2 = 4 \Rightarrow x = \pm 2 ]

6. Check the denominator condition:

   • (x=-3) is not a solution anyway, so no extra restriction removes the found roots.
7. Determine the x-intercepts:
   • For (x=2): ((2,0))
   • For (x=-2): ((-2,0))

Goal 2: Intersection with the y-axis

1. Set (x = 0) to find (y): [ y = \frac{0^2 - 4}{0+3} = \frac{-4}{3} ]

2. Determine the y-intercept: [ \left(0, -\frac{4}{3}\right) ]

Graphing instruction

• Plot the three intercept points found:
  • ((-2,0))
  • ((2,0))
  • ((0, -\frac{4}{3}))
• The graph should pass through these points.

Wrap-up / next topic

• The instructor says that the next video will analyze asymptotes of the function graph.
• Viewers are encouraged to subscribe and watch the next video.

Speakers / sources

• Math tutor / instructor (unnamed speaker; the “I” voiceover throughout the subtitles).