Lines, curves and coordinates
Introduced| English | Meaning |
|---|---|
| gradient/ˈɡreɪdɪənt/ | The change in the vertical coordinate divided by the change in the horizontal coordinate. |
| y-intercept/waɪ ˌɪntəˈsept/ | The point or coordinate where a graph meets the vertical y-axis. |
| parallel/ˈpærəlel/ | Describing lines in the same plane that never meet, however far extended. |
| perpendicular/ˌpɜːpənˈdɪkjʊlə/ | Meeting at a right angle of 90 degrees. |
| midpoint/ˈmɪdpɔɪnt/ | The point halfway between the endpoints of a line segment. |
| distance/ˈdɪstəns/ | The total length of a path travelled. |
| roots/ruːts/ | Values of a variable that make an equation or polynomial expression equal to zero. |
| parabola/pəˈræbələ/ | The curve formed by points equally distant from a fixed point and a fixed straight line. |
| vertex/ˈvɜːteks/ | A corner or meeting point of edges, a turning point of a parabola, or a node in a graph, depending on the context. |
| graphically/ˈɡræfɪkli/ | Using a graph or diagram to represent information or find a solution. |
Connect a rule with a graph
- Coordinates let you test where a line or curve passes. A picture suggests relationships; algebra checks exact values.
- Label axes and relevant points when sketching. A graph's scale affects what can be read accurately from it.
State which lines have a gradient
- In $y=mx+c$, m is the gradient 斜率 and c the y-intercept y 轴截距. For a nonvertical line, $m=(y_2-y_1)/(x_2-x_1)$.
- Distinct nonvertical parallel 平行 lines have equal gradients; nonvertical perpendicular 垂直 lines have gradient product negative 1. Vertical lines have undefined gradient and are perpendicular to horizontal lines.
Change the gradient and the intercept
$y = mx + c$
See what m and c each control before you calculate them.
Find the gradient of the line through (2, 1) and (6, 9).
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Rise over run: (9 − 1) ÷ (6 − 2) = 8 ÷ 4 = 2.
That line is y = 2x + c. Find c.
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Substitute (2, 1): 1 = 4 + c, so c = −3. Check with (6, 9): 2(6) − 3 = 9 ✓.
Which gradient is perpendicular to a line of gradient 4?
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The gradients must multiply to −1, and 4 × (−1/4) = −1.
Measure segments using coordinates
- The midpoint 中点 averages each coordinate: $M=((x_1+x_2)/2,(y_1+y_2)/2)$.
- The distance 距离 is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. A perpendicular bisector passes through the midpoint at a right angle to the segment.
Find the distance between (1, 2) and (4, 6).
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Differences 3 and 4, so d = √(9 + 16) = √25 = 5 — the 3-4-5 triangle.
Roots can cross or touch
- A quadratic graph is a parabola 抛物线; its vertex 顶点 is the turning point. Real roots 根 are x-values where it meets the x-axis, including a repeated root that touches.
- To solve graphically 用图像求解, read intersections of the relevant graphs. Equate their expressions to check exact coordinates and distinguish a strict inequality from equality at an endpoint.
What are the roots of a quadratic, geometrically?
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A real root satisfies y = 0. The graph can cross the x-axis or touch it at a repeated root.
Put the steps of finding a line through two points in order.
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Checking with the unused point is the step that catches an arithmetic slip.
Find a line from two points. For $(2,1)$ and $(6,9)$, $m=(9-1)/(6-2)=2$. In $y=mx+c$, $c=y-mx=1-2(2)=-3$. Thus $y=2x-3$; substituting the other point gives 9 as required.
Find both coordinates of an intersection. The equations $y=x+1$ and $y=x^2-3x+1$ give $x(x-4)=0$. Substitution gives $(0,1)$ and $(4,5)$, not just two x-values. Sheet 1.3 also asks for a labelled quadratic sketch.
Equal gradients alone do not distinguish identical from distinct parallel lines. The negative-reciprocal test does not cover a vertical line. Check the equation form before applying a rule.