Quadratic graphs

For each graph below, select its possible equation from this list.

A x = 4 B y = 4 C y = x

Write the letter of the equation beneath each graph.

The graph of the line y = 2 - is shown on this grid.

By drawing a suitable line on the grid, solve these simultaneous equations.

The graph of the line x + y = 3 is shown on this grid.

By drawing a suitable line on the grid, solve these simultaneous equations.

x + y = 3 and y = 2 x - 3

Umi says,

The simultaneous equations y = 2x + 3 and 2y – 4x = 7 will have no solutions.

Is Umi correct? Explain how you decide.

Write x² - 6x + 5 in the form (x - a)² - b.

Write down the coordinates of the turning point of y = x² - 6x + 5.

On the axes below, sketch the graph y =-x².

Write down a possible equation for the graph sketched below.

Sketch the graph of y = 4 on the axes below. Show clearly the value of any intercepts.

Show that (x + 3)(x - 5) = x² - 2x - 15.

The diagram shows a sketch of the graph y = (x + 3)(x - 5).

Complete the diagram by adding the values of the three intercepts with the axes.

The minimum point on the graph is marked T.

Write down the coordinates of the point T.

Here is a table of values for y = x² - x - 8.

Draw the graph of y = x² - x - 8 for -4 ⩽ x ⩽ 4.

Write down the equation of the line of symmetry of the graph.

Use the graph to solve the equation x² - x - 8 = 0. Give your answers to 1 decimal place.

Here is a table of values for y = 2x² - 3x.

Draw the graph of y = 2x² - 3x for values of x from 2^- to 3.

Use your graph to find the x-coordinates of the points where the graph of y = 2x² - 3x crosses the line y = 6.

(b) x =........ and x =........

Solve by factorisation.

Write x² + 8x + 11 in the form (x + a)² - b.

Write down the coordinates of the turning point of the graph y = (x - 3)² + 8.

Describe the single transformation which maps the graph of y = x² onto the graph of y = (x - 3)² + 8.

Work out the coordinates of the intersection of the graphs of y = 5 - 2x and y = 3x². You must show your working.

(........,........) and (........,........)

Complete this table for y = x³ - 3x².

Draw the graph of y = x³ - 3x² for values of x from 2^- to 4.

Use the graph to solve the equation x³ - 3x² = 5. Give your answer to 1 decimal place.

For each graph below, select its possible equation from this list.

A y = - 3 B y = x C y = x + 1

D y = x² E x = -3 F y = x³

Write the letter of the equation beneath each graph.

Here are sketches of six graphs, labelled A to F.

Write the letter of the graph that represents the following relationships.

y is directly proportional to x.

y is inversely proportional to x.

Complete the table for y = x³ + 4x².

Draw the graph of y = x³ + 4x² for -5 ⩽ x ⩽ 1.

The equation x³ + 4x² = k, where k is an integer, has exactly one solution for -5 ⩽ x ⩽ 1.

Find the greatest possible value of k.

The x-coordinates of the intersections of the graphs of y = x² + ax + 7 and y = 3x + b are the solutions to the equation x² + 8x + 13 = 0.

Find the value of a and the value of b.

Here is a table of values for y = - 2x.

Draw the graph of y = - 2x for 4 ⩽ x ⩽ 4, x ≠ 0.

Use your graph to find the positive solution of - 2x = 0. Give your answer to 1 decimal place.

Here is a table of values for y = - 2x.

Draw the graph of y = - 2x for 4 ⩽ x ⩽ 4, x ≠ 0.

Use your graph to find the positive solution of - 2x = 0. Give your answer to 1 decimal place.

The time taken to complete a journey halves as the speed doubles.

On the axes below, sketch a graph to show this relationship.

It takes 40 minutes to fill a garden pond using water from 5 identical hose pipes.

Assuming the rate of flow of water from each hose pipe is the same, work out how many minutes it would take to fill the same garden pond using 2 of these hose pipes.

The diagram shows the graph of y = kx - x² + 2, where k is an integer.

Show that k = 3.

Use the graph to solve 3x - x² + 2 = 3. Give your answers to 1 decimal place.

Write (2x - 5)(x + 4) in the form 2(x + a)² - b.

You must show your working.

Charlie, Dev and Eve all attempt to sketch the graph of y = (2 x - 5)(x + 4).

Charlie‛s sketch
Dev‛s sketch Eve‛s sketch

Whose sketch is the most accurate? Write down the properties of the graph that you used in making your decision.

Sketch the graph of y = - 2. Show clearly the value of any intercepts.

Sketch the graph of y = x - 3. Show clearly the value of any intercepts.

Sketch the graph of y = x².

The graph shows the solution to this pair of simultaneous equations.

2 x + y = 14 x + 2 y = 13

Use the solution to work out the value of 3 x + y. You must show how you work out your answer.

By factorising, find the roots of y = x² + 18x + 77.

(a) x =........ and x =........

Write y = x² + 18x + 77 in the form y = (x + a)² - b.

Write down the coordinates of the turning point of the graph of y = x² + 18x + 77.

The graph of y = is drawn on the grid for -2 ⩽ x ⩽ 6.

There are no values of x for which = k. Find the value of k.

Use the graph to find approximate solutions to the equation = 3x - 1. Give your answers to 1 decimal place. Show your working on the graph.

(b)(i) x =........ or x =........

Complete this table for y = x² - 5.

Draw the graph of y = x² - 5 for the values of x from -4 to 3.

Use the graph to solve the equation x² - 5 = 0. Give your answers to 1 decimal place.

This is a sketch of the graph of y = x² - 10x + 16.

Write down the value of the y-intercept.

Write down the x-coordinate of the turning point.

Here is a table of values for y = x² + 2x - 2.

Draw the graph of y = x² + 2x - 2 for -4 ⩽ x ⩽ 2.

Write down the equation of the line of symmetry of the graph.

Use the graph to solve the equation x² + 2x - 2 = 0. Give your answers to 1 decimal place.

(c) x =........ or x =........

The graph of y = x² + 6x - 2 is shown below. The roots of the equation x² + 6x - 2 = 0 are at p and q.

Calculate y when x = 1.

Without solving the equation, explain why q must lie between 0 and 1.

Explain why using a method of iteration is not the most appropriate way of finding a solution to this equation.

Frankie sketches this quadratic graph.

Frankie says

The y-intercept is 15.

Show that what Frankie says could be correct.

Explain why what Frankie says may not be correct.

For each graph below, select its possible equation from this list.

A y = x³ B y = -2 C y = -x

D x = -2 E y = x² F y = 2x + 1

Write the letter of the equation beneath each graph.

Here are sketches of five graphs.

Write the letter of the graph that represents the following relationships.

y is directly proportional to x.

y is inversely proportional to x.

Complete this table for y = x² - 3.

Draw the graph of y = x² - 3 for values of x from -3 to 3.

Use your graph to solve x² - 3 = 2.

A straight line has the equation y = 2 x - 1.

Write down the gradient of the line.

Here are the equations of four straight lines.

Which of the four straight lines is parallel to y = 2 x - 1 ?

A student says

y = x + 4 is the steepest of the four straight lines because it has the largest number added.

Explain why the student is wrong.

Here is part of the graph of y = 2 x - 1.

The line continues to the right.

Will the line pass above, below or through the point (45, 90)? Show how you decide.

The line y = 2 x - 1 will pass........ the point (45, 90) because........

Find the coordinates of the turning point of the graph of y = x² + 6x + 17.

For each graph below, select its possible equation from this list.

Here is a sketch of y = x³.

On the axes below, sketch the graphs of

y = -x³

y = x³ - 8, showing the values of any intercepts with the axes.

Here are sketches of five graphs.

Write the letter of the graph that represents the following relationships.

y is directly proportional to x.

y is inversely proportional to x.

Sketch the graph of x = 3. Show clearly the value of any intercepts.

Sketch the graph of y = x² + 1. Show clearly the value of any intercepts.

Toby has sketched the graph of y = below.

Make two comments about the accuracy of his sketch.

Complete this table for y = x² - x.

The graph of y = x + 1 is shown on the grid.

On the same grid, use part (a) to draw the graph of y = x² - x for values of x from -2 to 3.

Write down the x-coordinates of the points where y = x² - x and y = x + 1 cross.

(c) x =........ and x =........

Write x² - 10x + 22 in the form (x - a)² - b.

Sketch the graph of y = x² - 10x + 22. Show clearly the coordinates of any turning points and the value of the y-intercept.

The graph of y = 2x² + 3x - 9 is drawn below.

Use the graph to solve 2x² + 3x - 9 = 0.

The equation 2x² + x - 4 = 0 can be solved by finding the intersection of the graph of y = 2x² + 3x - 9 and the line y = ax + b.

Find the value of a and the value of b.

Hence use the graph to solve the equation 2x² + x - 4 = 0.

(ii) x =........ or x =........

Write x² - 6x + 11 in the form x - a² + b.

Sketch the graph of y = x² - 6x + 11. Show clearly the coordinates of any turning points.

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

Draw the graph of y = x² - 4x + 1 for -1 ⩽ x ⩽ 5.

On the same grid, draw the graph of y = 2 x - 6 for - 1 ⩽ x ⩽ 5.

Use your graphs to solve the equation x² - 4x + 1 = 2x - 6.

Give your answers to 1 decimal place.

Write x² + 8x + 3 in the form x + a² - b.

Sketch the graph of y = x² + 8x + 3. Show clearly the coordinates of any turning points and the y-intercept.

The graph of y = x³ - x² - 2 is drawn on the grid.

Use the graph to solve x³ - x² - 2 = 0. Give your answer correct to 1 decimal place.

The equation x³ - x² + 5x - 6 = 0 can be solved by finding the intersection of the graph of y = x³ - x² - 2 and the line y = ax + b.

Find the value of a and the value of b.

Hence, use the graph to solve the equation x³ - x² + 5x - 6 = 0. Give your answer correct to 1 decimal place.

Write down the equation of each of these lines.

Sketch the graph of y = x².

Here is the graph of y = x² + x - 6.

Use the graph to solve the equation x² + x - 6 = 0.

Write x² + 4x - 16 in the form (x + a)² - b.

Solve the equation x² + 4x - 16 = 0. Give your answers in surd form as simply as possible.

(ii) x =........ or x =........

Sketch the graph of y = x² + 4x - 16, showing clearly the coordinates of any turning points.

These graphs show different relationships between the variables x and y.

Identify the graph which shows the following.

y is directly proportional to x.

y is inversely proportional to x.

Sketch a graph which shows that y is directly proportional to x².

Write x² – 8x + 25 in the form (x – a)² + b.

Write down the coordinates of the turning point of the graph of y = x² – 8x + 25.

Hence describe the single transformation which maps the graph of y = x² onto the graph of y = x² – 8x + 25.

For each graph below, select its possible equation from this list.

Complete the table for y = x³ - 3x.

Draw the graph of y = x³ - 3x for -3 ⩽ x ⩽ 3.

Use your graph to solve x³ - 3x = 10.

Complete this table for y = x² - 5.

On the grid below, draw the graph of y = x² - 5 for the values of x from -2 to 4.

On the same grid, draw the line y = - 2.

Write down the x coordinates of the points where y = x² - 5 and y = -2 cross.

(d) x =........ and x =........

Complete this table for y = x² + x - 4.

Draw the graph of y = x² + x - 4 for -4 ⩽ x ⩽ 3.

Use your graph to solve x² + x - 4 = 0.

On the same grid, draw the graph of y = - 2 x - 1 for - 4 ⩽ x ⩽ 3. You may use the table if you wish.

Use your graphs to solve the equation x² + x - 4 = -2x - 1.

Write x² - 6x + 20 in the form (x - a)² + b.

Write down the turning point of the graph of y = x² - 6x + 20.

Complete the table for y = x² - 2x.

Draw the graph of y = x² - 2x for -1 ⩽ x ⩽ 4.

Use your graph to solve x² - 2x = 2.

Sketch the graph of y = x - 2² - 3. Show the coordinates of any turning points.

The sketch shows part of a graph which has equation y = ax² + bx + c.

Find the values of a, b and c.

Complete the table for y = x² - 2x.

Draw the graph of y = x² - 2x for -1 ⩽ x ⩽ 4.

Use your graph to solve x² - 2x = 2.