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Solved problems for the chapter:

Completing the square method

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Quadratic function

1. Formats of the quadratic function

1.1. General form and meaning of the leading coefficient a

1.2. Zero shape and zero square functions

1.3. Top shape and quadratic function themes

1.3.1. Specify the peak shape

1.3.2. Determining the apex shape by completing it to a perfect square

2. Draw a graph of a quadratic function (parabola)

2.1. Drawing a parabola based on a general shape

2.2. Drawing a parabola based on the vertex shape

3. Mutual position of parabola and line

4. Forum

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Vieta's formula

1. Derivation of Vieta's formula

2. Forum

↓ Show theory [2] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:completing_the_square_method/5386/

Example 1.

Given the quadratic function f(x) with the rule .

Calculate the zeros of the given function, the vertex point and draw the graph.
Find the range of values of x for which .
On what interval is the function increasing? Write down the largest of such interval.
Solve the equation .

Example 2.

Write down the factored and vertex forms of the quadratic function:

Given the quadratic function :

Find a real number b so that the line is the symmetric axis of the graph of the function f(x).
Write down the coordinates of the vertex.

Find the expression for the quadratic function whose vertex is on the abscissa axis, its curve passes through the point (- 3, - 3) and intersects the ordinate axis at - 48.

Given the function , where with the vertex at point (2, 6), find the value of u and v, and hence the function f(x).

Write the coordinates of the vertex of the quadratic function .
For m = 4, write the general and vertex form of the quadratic function.

Find the values of m for which the vertex of the parabola lies on the line .

Write a rule for the quadratic function that has its vertex on the ordinate axis and its graph goes through points (1, 5) and (- 2, 11).

Write the functions below in vertex form and draw their graphs.

10.

Determine the parameter m so that the vertex of the parabola lies on the line .

11.

Given the function .

Write the function in vertex form.
Draw graphs of the functions f(x) and in the same coordinate system.

12.

Write the equation of the quadratic function whose vertex is at on point and one of its zeros is .

13.

Given the following functions:

Calculate the zeros and vertex coordinates of function and accurately draw graphs of both functions in the same coordinate system.
Calculate the coordinates of the intersections of the graphs of the functions and .
Calculate the distance between the intersections. Leave your answer in surd form.

14.

The graph of the quadratic function f with zeros at and intersects the ordinate axis at the point . Write a rule for the f function, calculate the coordinates of the vertex, and draw its graph.