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Further Graphs

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선생님을 위해: Further Graphs(Maths [CIE], Core)을(를) 위한 바로 쓸 수 있는 수업 슬라이드, 복습 노트, 다이어그램 — 수업에 사용하거나, 학생들이 실시간 게임으로 즐기는 인터랙티브 클래스 활동으로 진행하세요.

수업 노트

Types of Graphs

  • Straight lines: y = mx + c (e.g., y = 3x + 2). Key lines: y = x and y = -x.
  • Horizontal lines: y = c (e.g., y = 4).
  • Vertical lines: x = k (e.g., x = 2).
  • Quadratic graphs: y = ax² + bx + c. Shape is a parabola: u-shaped (positive a) or n-shaped (negative a).
  • Reciprocal graphs: y = a/x. Two L-shaped branches; x ≠ 0. Positive a gives branches in first and third quadrants.

Positive and negative quadratics

Positive and negative quadraticsO−8−6−4−22468−4−3−2−11234xyMax (0, 2)y = −x² +2 (a < 0)y = x² −2 (a > 0)Min (0, −2)

Key Features of Quadratic Graphs

  • The turning point is called the vertex: minimum for positive quadratics, maximum for negative quadratics.
  • Quadratic graphs have a vertical line of symmetry through the vertex: x = k (k is x-coordinate of vertex).
  • Roots are x-intercepts where y = 0. A quadratic can have 0, 1 (touches), or 2 roots.
  • Roots are symmetric about the line of symmetry.
  • Quadratic graphs always have one y-intercept (where x = 0).

Key features of a quadratic graph

Key features of y = x² − 2x − 3O−4−2246−3−2−112345xy(−1, 0)(0, −3)(3, 0)y = x² −2x − 3Min (1, −4)

Drawing Graphs from Tables

  • Substitute x-values into the equation to find y-values. Use brackets for negative x and follow BIDMAS.
  • For reciprocal graphs, do not include x = 0 (division by zero).
  • Plot points accurately (within half a square). Join with a smooth freehand curve (no ruler).
  • Use calculator table function: enter function, start/end x, step size. Check given y-values.
  • If a point doesn't fit the curve shape, check your working.

Solving Equations from Graphs

  • To solve f(x) = 0, read the x-intercepts (roots) of the graph of y = f(x).
  • To solve f(x) = k, draw horizontal line y = k and read x-coordinates of intersections.
  • To solve f(x) = g(x), plot y = f(x) and y = g(x); solutions are x-coordinates of intersection points.
  • For equations not in the form 'graph = ...', rearrange to match the given graph. E.g., x² - 4x + 3 = 1 becomes x² - 4x - 2 = -4.
  • When solving for x, give only x-coordinates. Include y-coordinates only for simultaneous equations.

Solving f(x) = g(x) graphically

Solving x² + 3x + 1 = 2x + 1 graphicallyO−2−112345−4−3−2−112xyy = x² + 3x + 1y = 2x + 1(0, 1)(−1, −1)

Example: Quadratic Graph from Table

  • Substitute each x-value into y = x² − 4x − 3 to complete the table of values: | x | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---|---|---| | y | 9 | 2 | −3 | −6 | −7 | −6 | −3 | 2 |
  • Plot the points and draw a smooth u-shaped curve for −2 ≤ x ≤ 5.
  • Solve x² − 4x − 3 = 0 by reading the x-intercepts from the graph: x ≈ −0.65 and x ≈ 4.65.
  • Line of symmetry: x = 2, the midpoint of the roots (and −b/(2a) = 4/2 = 2).

y = x² − 4x − 3

y = x² − 4x − 3O−5510−2−112345xy(−0.65, 0)(4.65, 0)y = x² −4x − 3

Example: Reciprocal Graph

  • Substitute each x-value into y = 15/x to complete the table of values: | x | −5 | −3 | −2 | −1 | 1 | 2 | 3 | 5 | |---|---|---|---|---|---|---|---|---| | y | −3 | −5 | −7.5 | −15 | 15 | 7.5 | 5 | 3 |
  • Draw two separate branches, one for x < 0 and one for x > 0; do not connect them at x = 0.
  • Solve 15/x = 6 by drawing the horizontal line y = 6 and reading the x-coordinate of the intersection: x = 2.5.
  • Reciprocal graphs have asymptotes at x = 0 and y = 0.

y = 15/x

y = 15/x0510152025303512345678xy(2.5, 6)y = 15/x

Example: Solving with Horizontal Line

  • Substitute each x-value into y = 1 + 5x − x² to complete the table of values: | x | −1 | 0 | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---|---|---| | y | −5 | 1 | 5 | 7 | 7 | 5 | 1 |
  • Draw the horizontal line y = 3. The x-coordinates of the intersections solve 1 + 5x − x² = 3.
  • Solutions: approximately x = 0.44 and x = 4.56.
  • Line of symmetry: x = 2.5 (the midpoint of the two solutions above, and the x-coordinate of the vertex).

y = 1 + 5x − x²

y = 1 + 5x − x²O−6−4−22468−112345xy(0.44, 3)y = 1 +5x − x²y = 3(4.56, 3)

Example: Solving with Another Line

  • Substitute each x-value into y = −x² + x + 5 to complete the table of values: | x | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 | |---|---|---|---|---|---|---|---|---| | y | −7 | −1 | 3 | 5 | 5 | 3 | −1 | −7 |
  • Draw the line y = x. The x-coordinates of the intersections solve −x² + x + 5 = x, which simplifies to x² = 5.
  • Solutions: x = ±√5 ≈ ±2.24.
  • Vertex (turning point): (0.5, 5.25). Line of symmetry: x = 0.5.
  • Roots of −x² + x + 5 = 0: approximately x = −1.79 and x = 2.79.

y = −x² + x + 5

y = −x² + x + 5O−8−6−4−2246−3−2−11234xy(−2.24, −2.24)y = −x² +x + 5(2.24,2.24)y = x

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연습 문제

무료 미리 보기 — 57개 중 8개 문제. 가입하면 전부 볼 수 있어요.

  1. 1.Which of the following is the equation of a straight line?

    Easy
    • Ay = 3x + 2
    • By = x2 + 3x + 2
    • Cy = 1/x
    • Dy = -x2 + 3x + 2

  2. 2.What is the name of the shape of a quadratic graph?

    Easy
    • AParabola
    • BHyperbola
    • CStraight line
    • DCircle

  3. 3.For a quadratic graph y = ax2 + bx + c, if a is positive, the graph is:

    Easy
    • Au-shaped
    • Bn-shaped
    • Ca straight line
    • DL-shaped

  4. 4.What is the equation of the line of symmetry for a quadratic graph with vertex at (2, 5)?

    Easy
    • Ax = 2
    • By = 2
    • Cx = 5
    • Dy = 5

  5. 5.Which of the following is a reciprocal graph?

    Easy
    • Ay = 4/x
    • By = 4x + 1
    • Cy = x2 - 4
    • Dy = -x2 + 4

  6. 6.What is the value of y when x = -2 in the equation y = x2 - 3x?

    Easy
    • A10
    • B-2
    • C-10
    • D2

  7. 7.Which of the following is the equation of a horizontal line?

    Easy
    • Ay = 4
    • Bx = 4
    • Cy = x
    • Dy = -x

  8. 8.The graph of y = -x2 + 2x + 3 is:

    Easy
    • An-shaped
    • Bu-shaped
    • Ca straight line
    • DL-shaped

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