Theory of Intelligence
11TH•Mathematics•Functions And Graphs•Exercise 2.2

2.2 Finding the Intersecting Point(s) Graphically

Practice set 2.2 for Functions and Graphs (intercepts, intersections, graphing, applications).

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Q1

Question 1

Find the point of intersection of the coordinate axes and the following linear functions graphically:

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(i)

y=−5x+10y=-5x+10

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Step-by-Step Solution

Question 1 • Part (i)

verifiedVerified Answer

(i) y=−5x+10y = -5x + 10

Find the point of intersection of the coordinate axes and the linear function graphically.

polyline
Graphing Lab•Interactive

Interactive Coordinate Axes & Intercepts

polylineInteractive Graphing Lab • Section 2.2

Axes Intercepts: y = -5x + 10

Steep negative slope intersecting at (2, 0) and (0, 10)

y = -5x + 10
Intersection / Intercept Points
xy-101234-224681012x-intercept (2, 0)y-intercept (0, 10)
2 Key Feature Points Detected
xy = -5x + 10Status
-0.512.5Regular Point
010Axis Intercept
15Regular Point
20Axis Intercept
2.5-2.5Regular Point
lightbulbCore Principle of Intersection:

A point $(x, y)$ is on a graph if and only if its coordinates satisfy the equation. Therefore, an intersection between two curves means both rules output the same $y$ for the exact same input $x$ ($f(x) = g(x)$). Graphically, it is the meeting point; algebraically, it is the simultaneous solution!

Step-by-Step Solution

  1. Find the yy-intercept (set x=0x = 0):

    y=−5(0)+10=0+10=10\begin{aligned} y &= -5(0) + 10 \\ &= 0 + 10 = 10 \end{aligned}

    The line crosses the yy-axis at (0,10)(0, 10).

  2. Find the xx-intercept (set y=0y = 0):

    0=−5x+105x=10x=105=2\begin{aligned} 0 &= -5x + 10 \\ 5x &= 10 \\ x &= \frac{10}{5} = 2 \end{aligned}

    The line crosses the xx-axis at (2,0)(2, 0).

Table of Values

xx−0.5-0.50011222.52.5
y=−5x+10y = -5x + 1012.512.51010 (yy-int)5500 (xx-int)−2.5-2.5
x-intercept: (2,0),y-intercept: (0,10)\boxed{\text{$x$-intercept: }(2, 0), \quad \text{$y$-intercept: }(0, 10)}
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