Roots and turning points of quadratic graphs

The curve C has equation y = (x − 3)² + 5

Write down the coordinates of the turning point of C.

The curve y = x² - 6x + 8 crosses the x-axis at x = 2 and at x = 4.

1. Write down the solutions of x² - 6x + 8 = 0
2. Write down the x-coordinate of the turning point of the curve.

A quadratic graph has the equation y = (x − 2)(x − 6).

1. Write down the coordinates of the points where the graph crosses the x-axis.
2. Work out the coordinates of the point where the graph crosses the y-axis.
3. Sketch the graph, showing these points.

A quadratic graph has equation y = (x + 3)(x − 7).

1. Write down the roots of the equation.
2. Write down the equation of the line of symmetry of the graph.
3. Work out the coordinates of the turning point.

The diagram shows the graph of y = x² - 2x - 3

1. Write down the coordinates of the point where the graph crosses the y-axis.
2. Write down the solutions of x² - 2x - 3 = 0
3. Write down the coordinates of the turning point of the graph.

The graph of y = x² is translated 3 units to the right and 2 units down to give a new graph.

1. Write down the equation of the new graph in the form y = (x − a)² + b.
2. Write down the coordinates of the turning point of the new graph.
3. Work out where the new graph crosses the y-axis.

y = x² − 8x + 7

1. Write x² − 8x + 7 in the form (x + a)² + b.
2. Write down the coordinates of the turning point and the equation of the line of symmetry of the graph.
3. Work out the coordinates of the points where the graph crosses the x-axis.

By completing the square, find the coordinates of the minimum point of the curve

y = x² + 10x − 2

y = −x² + 6x − 5

1. Write y in the form y = −(x − p)² + q.
2. Write down the coordinates of the turning point and say whether it is a maximum or a minimum.
3. Sketch the graph, showing where it crosses the axes.

1. Complete the table of values for y = 6 - x²

   x	-3	-2	-1	0	1	2	3
   y	-3		5		5		-3

2. Write down the coordinates of the maximum point of the graph of y = 6 - x²
3. Use the table to write down the two consecutive whole numbers that the positive solution of 6 - x² = 0 lies between.

The graph of y = x² + bx + c passes through the points (0, 8) and (6, 8).

1. Write down the equation of the line of symmetry of the graph.
2. Find the values of b and c.
3. Work out the coordinates of the turning point.

1. Complete the table of values for y = x² + 2x - 8

   x	-4	-3	-2	-1	0	1	2
   y	0	-5		-9		-5

2. Use the table to write down the solutions of x² + 2x - 8 = 0
3. Write down the equation of the line of symmetry of the graph of y = x² + 2x - 8

1. Complete the table of values for y = x² - 4x

   x	-1	0	1	2	3	4	5
   y	5	0			-3		5

2. On the grid, draw the graph of y = x² - 4x for values of x from -1 to 5
3. Write down the coordinates of the turning point of the graph.

1. Complete the table of values for y = x² - 8x + 11

   x	0	1	2	3	4	5	6	7	8
   y	11	4		-4		-4		4	11

2. Write down the coordinates of the turning point of the graph of y = x² - 8x + 11
3. Write down the equation of the line of symmetry of the graph.

The curve C has equation y = x² − 4x − 5

1. Write x² − 4x − 5 in the form (x − a)² + b

2. Sketch C, showing clearly the coordinates of the turning point and the coordinates of the points where C crosses the axes.

1. Complete the table of values for y = 2x² - 3

   x	-2	-1	0	1	2
   y	5			-1

2. Write down the coordinates of the turning point of the graph of y = 2x² - 3
3. Use the table to write down the two consecutive whole numbers that the positive solution of 2x² - 3 = 0 lies between.

1. Complete the table of values for y = x² + x - 6

   x	-4	-3	-2	-1	0	1	2	3
   y	6	0		-6		-4		6

2. On the grid, draw the graph of y = x² + x - 6 for values of x from -4 to 3
3. Write down the solutions of x² + x - 6 = 0
4. Use your graph to estimate the solutions of x² + x - 6 = -5