0580/33 May/Jun 2023, Question 6: Quadratic Graphs and the Parabola

Part (a) · 2 marks

Find the equation of the straight line L shown on the grid, giving it as y = mx + c.

1. 1L crosses the y-axis at (0, 3), so c = 3
2. 2L passes through (0, 3) and (1, 1): gradient = (1 − 3) ÷ (1 − 0) = −2
3. 3y = −2x + 3

Answer−2x + 3

Part (bi) · 2 marks

Fill in the missing y-values in the table for y = x² − 3x − 3 at the given x-values.

1. 1x = −2: 4 + 6 − 3 = 7
2. 2x = 0: −3
3. 3x = 3: 9 − 9 − 3 = −3
4. 4x = 5: 25 − 15 − 3 = 7

Answer7, −3, −3, 7

Part (bii) · 4 marks

Plot and draw the curve y = x² − 3x − 3 on the grid for x from −2 to 5.

1. 1Plot (−2, 7), (−1, 1), (0, −3), (1, −5), (2, −5), (3, −3), (4, 1), (5, 7)
2. 2Join with a smooth U-shaped curve, lowest point between x = 1 and x = 2

AnswerCorrect and accurate curve

Part (ci) · 1 mark

State the coordinates of the minimum point of the curve y = x² − 3x − 3.

1. 1By symmetry the lowest point is at x = 1.5
2. 2y = 1.5² − 3 × 1.5 − 3 = −5.25
3. 3(1.5, −5.25)

Answer(1.4 to 1.6, −5.5 to −5)

Part (cii) · 1 mark

Draw the vertical line of symmetry of the curve on the grid.

1. 1Draw a vertical ruled line through x = 1.5

AnswerLine x = 1.5 drawn accurately

Part (ciii) · 1 mark

Give the equation of the line of symmetry of y = x² − 3x − 3.

1. 1From the table, y = −5 at x = 1 and at x = 2
2. 2The line of symmetry is halfway: (1 + 2) ÷ 2 = 1.5
3. 3x = 1.5

Answerx = 1.5

Part (d) · 1 mark

Find the coordinates where line L crosses the curve y = x² − 3x − 3 at positive x.

1. 1−2x + 3 = x² − 3x − 3
2. 2x² − x − 6 = 0, (x − 3)(x + 2) = 0
3. 3For x > 0: x = 3, y = −2 × 3 + 3 = −3
4. 4(3, −3)

Answer(3, −3)

More past-paper questions on quadratic graphs and the parabola

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