Tangent Line Through Point C
A tangent line is a line that intersects a curve at precisely one point, with the line's slope matching the instantaneous slope of the curve at the point of contact. In the provided graph, the curve exhibits a constant slope of 5. Consequently, the tangent line traversing point C will also possess a slope of 5.
To establish the equation for the tangent line passing through point C, we can leverage the point-slope form of linear equations given by:
y - y₁ = m(x - x₁)
where 'm' represents the slope, and '(x₁, y₁)' signifies the point of intersection.
Given that point C has coordinates (2, -3) and the slope (m) is 5, we can substitute these values into the equation:
y - (-3) = 5(x - 2)
y + 3 = 5x - 10
y = 5x - 13
Therefore, the equation of the tangent line traversing point C is y = 5x - 13.
Secant Line Through Points B and F
A secant line intersects a curve at two distinct points. Since points B and F reside on the curve, a line drawn through them will intersect the curve at those specific points.
To determine the equation of the secant line connecting points B and F, we can employ the two-point form of linear equations:
y - y₁ = (y₂ - y₁)/(x₂ - x₁) (x - x₁)
where '(x₁, y₁)' and '(x₂, y₂)' represent the two points.
Given that B's coordinates are (0, -5) and F's coordinates are (5, 5), we can substitute these values into the equation:
y - (-5) = (5 - (-5))/(5 - 0) (x - 0)
y + 5 = 10/5 (x - 0)
y + 5 = 2x
y = 2x - 5
Therefore, the equation of the secant line traversing points B and F is y = 2x - 5.
Tangent Lines Touching Point C
As previously established, a tangent line makes contact with a curve at a single point, and the slopes of the tangent line and curve are equivalent at the point of contact. Given the curve's constant slope of 5, any line with a slope of 5 that intersects the curve at point C qualifies as a tangent line.
There exist infinitely many tangent lines that can satisfy these conditions. Here are two examples:
The previously derived tangent line with the equation y = 5x - 13 satisfies the criteria.
Another tangent line could possess a distinct y-intercept while maintaining the slope of 5. For instance, the equation y = 5x + 7 would also represent a tangent line intersecting point C.
Secant Lines and Intersection Points
Ten secant lines would intersect the graph at various points. Determining the equations and intersection points for each of the ten lines would be rather extensive. However, we can discuss the general procedure for finding them.
To draw a secant line intersecting the graph at two distinct points, follow these steps:
Choose two separate points on the curve. These points can be located anywhere along the curve.
Employ the two-point form of linear equations, as described earlier, to establish the equation of the secant line.
The secant line will intersect the curve at the two points you selected in step 1.
Additional Considerations
The graph appears to depict a line, which by definition extends infinitely in both directions. Consequently, it would be possible to construct infinitely many secant lines that intersect the graph.
Due to the constant slope of the line, any two points on the line will result in the same slope when connected. Therefore, all secant lines drawn on this graph will also exhibit a slope of 5.