Concept explainers
- (a) What is the degree of a quadratic function f? What is the standard form of a quadratic function? How do you put a quadratic function into standard form?
- (b) The quadratic function f(x) = a(x − h)2 + k is in standard form. The graph of f is a parabola. What is the vertex of the graph of f? How do you determine whether f(h) = k is a minimum or a maximum value?
- (c) Express f(x) = x2 + 4x + 1 in standard form. Find the vertex of the graph and the maximum or minimum value of f.
(a)
To find: The degree of quadratic function f and its standard form and the method to put a quadratic function into standard form.
Answer to Problem 1RCC
The degree of the quadratic function
Explanation of Solution
Calculation:
The general form of a quadratic function is,
Where, a, b and c represents a real number.
And the degree of the quadratic function
That is, quadratic function f is a polynomial function of second degree.
The standard form of a quadratic function
And the parabola opens upward if
Quadratic expression can be expressed in standard form by completing the square.
Assume a quadratic function
Convert the quadratic function
Simplify further as follows,
Thus, the quadratic function
Therefore, the degree of the quadratic function
(b)
To explain: The vertex of the graph
Explanation of Solution
The given function is,
It is in the form of parabola.
Thus, the vertex of the parabola is
The parabola opens upward if
Thus, if
(c)
To find: The standard form of
Answer to Problem 1RCC
The standard form of the function
The maximum value of the expression f is
Explanation of Solution
Given:
The given equation is,
Formula used:
Maximum or minimum value of a quadratic function:
The standard form of a quadratic function
If
If
Calculation:
The quadratic expression can be expressed in standard form by completing the square.
Convert the given quadratic function in standard form as follows,
Thus, the quadratic function in standard form is
Compare this equation with the standard form
Where,
The vertex of the parabola is
So the vertex of the graph of the given function is
From the above formula, it is known that minimum value exists at
The quadratic function
Therefore, minimum value of the function
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Chapter 3 Solutions
Precalculus: Mathematics for Calculus (Standalone Book)
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