Quadratics and Curve Analysis
Explore various scenarios involving quadratics and curves, such as factorization, point on the curve, symmetry, vertex tracing, and minimum values. Discover how different values impact the properties of the curves and roots, with practical examples and solutions provided along the way.
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(1) (2) (3) What values could fill the gap so that the quadratic factorises nicely? What must fill the gap if the point (2,15) lies on the curve? Fill in the gap if the graph is symmetrical about the ?-axis. (8) (4) Quadratics 1 As the value in the gap changes, what curve does the parabola s vertex trace out? Fill the gap in two different ways to make the minimum value of ? equal 13. ? = ?2+ ? 9 (7) (6) (5) What point always lies on the curve, whatever number fills in the gap? What could fill the gap if the difference between the two roots is 6.5? Fill the gap in two different ways to make ? = ? 10 a tangent to the curve.
(1) (2) (3) What values could fill the gap so that the quadratic factorises nicely? ? or ? or ? What must fill the gap if the point (2,15) lies on the curve? ?? Fill in the gap if the graph is symmetrical about the ?-axis. ? (8) (4) Quadratics 1 As the value in the gap changes, what curve does the parabola s vertex trace out? Fill the gap in two different ways to make the minimum value of ? equal 13. ? or ? ? = ?2+ ? 9 Answers! ? = ?? ? (7) (6) (5) What point always lies on the curve, whatever number fills in the gap? What could fill the gap if the difference between the two roots is 6.5? 2.5 2.5 or 2.5 Fill the gap in two different ways to make ? = ? 10 a tangent to the curve. ? or ? 2.5 (?, ?)
(1) (2) (3) Fill the gap so that ? 0 for all values of ?. Find the minimum value of ? if the curve goes through the point (11,12). Fill the gap so that the ?-intercept of the curve is at (0,36). (8) (4) Quadratics 2 Fill the gap if the graph has equation ? = ?2 4? after being translated by the vector 4 What values could go in the gap if the line ? = 1 intersects the curve twice? ? = (? )(? 9) 5. (7) (6) (5) Find two ways to fill the gap if the line ? + ? = 5 is a tangent to the curve. Fill the gap so that the curve has the same line of symmetry as ? = ?2 15? + 60. Fill the gap so that the curve has the same roots as ? = 4?2 41? + 45.
(1) (2) (3) Fill the gap so that ? 0 for all values of ?. Find the minimum value of ? if the curve goes through the point (11,12). ? Fill the gap so that the ?-intercept of the curve is at (0,36). ? ? (8) (4) Quadratics 2 Fill the gap if the graph has equation ? = ?2 4? after being translated by the vector 4 What values could go in the gap if the line ? = 1 does not intersect the curve? ? < ? = (? )(? 9) 5. Answers! ? < ?? (7) (6) (5) Find two ways to fill the gap if the line ? + ? = 5 is a tangent to the curve. ? or ?? Fill the gap so that the curve has the same line of symmetry as ? = ?2 15? + 60. ? Fill the gap so that the curve has the same roots as ? = 4?2 41? + 45. 1.25 1.25
(1) (2) (3) Find two numbers to fill the gap if the point (12,7) lies on the curve. Find two numbers to fill the gap if one of the roots of the curve is 3. What is the minimum value of ?? Explain your answer. (8) (4) Quadratics 3 Show that the difference between the two roots of the curve is always 6. What values could fill the gap so that the curve has two negative roots? 2 9 ? = ? (7) (6) (5) Find two numbers to fill the gap if the vertex of the curve is 15 units from the origin. Fill the gap if the reflection of the curve in the line ? = 5 has equation ? = ?2 6?. What must fill the gap if the line ? = 18? intersects the curve at its vertex?
(1) (2) (3) Find two numbers to fill the gap if the point (12,7) lies on the curve. ? or ?? Find two numbers to fill the gap if one of the roots of the curve is 3. ? or ? What is the minimum value of ?? Explain your answer. 2 0 ?, e.g. because ? (8) (4) Quadratics 3 Show that the difference between the two roots of the curve is always 6. What values could fill the gap so that the curve has two negative roots? < ? 2 9 ? = ? Answers! (many different methods possible) (7) (6) (5) Find two numbers to fill the gap if the vertex of the curve is 15 units from the origin. ?? or ?? Fill the gap if the reflection of the curve in the line ? = 5 has equation ? = ?2 6?. ? What must fill the gap if the line ? = 18? intersects the curve at its vertex? 0.5 0.5
