Mathematics — first biennium · Lesson
The form y−y₀=k(x−x₀) describes all non-vertical lines through a fixed point P(x₀,y₀). Parameter k selects the slope of one member.
Prerequisites: Line through two points and equation forms
Goal: Interpret y−y₀=k(x−x₀) as a family of lines through a fixed point and determine one member from a condition.
In this lesson: Interpret y−y₀=k(x−x₀) as a family of lines through a fixed point and determine one member from a condition.
New concept: A family of lines contains a parameter: each parameter value gives a different line. A fixed point belongs to every line in the family when its coordinates satisfy the equation for every parameter value.
How to proceed: Substitute Q into the family equation: one equation in parameter k remains. Do not treat k as a coordinate.
You can already assume: Line through two points and equation forms.
The graph shows one member of the family: changing parameter k changes the slope, but every line in the family passes through P.
Completed exercises: 0/5 · Correct on first attempt: 0
Substitute Q into the family equation: one equation in parameter k remains. Do not treat k as a coordinate.
Data / situation: Consider the family y−(-1)=k[x−(-1)]. Which value of k gives the line also passing through Q(2,-7)?
ConceptStep separates what is required from what is recommended or useful as reinforcement. Only required prerequisites block your progress.
Relationships: solid line = required · dashed line = recommended · dotted line = reinforcement.