Problem 16
Question
For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ f(x)=-x^{2}+4 x+3 $$
Step-by-Step Solution
Verified Answer
The function has a maximum value of 7, and the axis of symmetry is x = 2.
1Step 1: Identify the Type of Quadratic
The quadratic function given is \( f(x) = -x^2 + 4x + 3 \). This function is in the standard form \( ax^2 + bx + c \), where \( a = -1 \), \( b = 4 \), and \( c = 3 \). The coefficient \( a = -1 \) is negative, indicating that the parabola opens downward. Thus, the function has a maximum value.
2Step 2: Calculate the Axis of Symmetry
The axis of symmetry for a quadratic function \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Plug \( b = 4 \) and \( a = -1 \) into the formula:\[x = -\frac{4}{2(-1)} = \frac{4}{2} = 2\]Thus, the axis of symmetry is \( x = 2 \).
3Step 3: Find the Maximum Value
Substitute \( x = 2 \) back into the original function to find the maximum value:\[f(2) = -2^2 + 4 imes 2 + 3 = -4 + 8 + 3 = 7\]Thus, the maximum value of the function is 7.
Key Concepts
Axis of SymmetryMaximum and Minimum ValuesStandard Form of Quadratic Equation
Axis of Symmetry
The axis of symmetry is a vertical line that divides a parabola into two mirror-image halves. In a quadratic function of the form \( ax^2 + bx + c \), the axis of symmetry can be determined using the formula \( x = -\frac{b}{2a} \). This formula helps us to find the vertical line where the parabola is evenly split.
In our original exercise, we have \( a = -1 \) and \( b = 4 \). This leads to the calculation:
\[x = -\frac{4}{2 \times (-1)} = \frac{4}{2} = 2\]
This tells us that the axis of symmetry is at \( x = 2 \). In simpler terms, if you were looking at a graph of this function, you would draw a straight vertical line at \( x = 2 \) running through the peak (or lowest point, if the parabola faced upwards) of the parabola.
Understanding the axis of symmetry can help in graphing the function and predicting the behavior of the quadratic equation.
In our original exercise, we have \( a = -1 \) and \( b = 4 \). This leads to the calculation:
\[x = -\frac{4}{2 \times (-1)} = \frac{4}{2} = 2\]
This tells us that the axis of symmetry is at \( x = 2 \). In simpler terms, if you were looking at a graph of this function, you would draw a straight vertical line at \( x = 2 \) running through the peak (or lowest point, if the parabola faced upwards) of the parabola.
Understanding the axis of symmetry can help in graphing the function and predicting the behavior of the quadratic equation.
Maximum and Minimum Values
When working with quadratic functions, it's important to determine whether the function has a maximum or minimum value. This is dictated by the direction in which the parabola opens. If the parabola opens upwards, it has a minimum value. If it opens downwards, it has a maximum value. The sign of the coefficient \( a \) in the standard form \( ax^2 + bx + c \) reveals this.
In our example, \( a = -1 \). Since it is negative, the parabola opens downward. This tells us that the quadratic function has a maximum value.
To find the maximum value, use the axis of symmetry \( x = 2 \) and substitute it into the original function \( f(x) = -x^2 + 4x + 3 \):
\[f(2) = -2^2 + 4 \times 2 + 3 = -4 + 8 + 3 = 7\]
Thus, the maximum value of the function is 7. This peak value of 7 occurs at \( x = 2 \), which provides a critical insight into the highest point the function reaches.
In our example, \( a = -1 \). Since it is negative, the parabola opens downward. This tells us that the quadratic function has a maximum value.
To find the maximum value, use the axis of symmetry \( x = 2 \) and substitute it into the original function \( f(x) = -x^2 + 4x + 3 \):
\[f(2) = -2^2 + 4 \times 2 + 3 = -4 + 8 + 3 = 7\]
Thus, the maximum value of the function is 7. This peak value of 7 occurs at \( x = 2 \), which provides a critical insight into the highest point the function reaches.
Standard Form of Quadratic Equation
The standard form of a quadratic equation is \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). This form makes it straightforward to identify the coefficients needed for calculating both the axis of symmetry and the maximum or minimum values.
In this specific exercise, the function is \( f(x) = -x^2 + 4x + 3 \), which is already presented in standard form. We observe the coefficients:
In this specific exercise, the function is \( f(x) = -x^2 + 4x + 3 \), which is already presented in standard form. We observe the coefficients:
- \( a = -1 \), which influences the direction of the parabola (upward or downward).
- \( b = 4 \), which is used to find the axis of symmetry.
- \( c = 3 \), which is the y-intercept of the quadratic function.
Other exercises in this chapter
Problem 16
For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ f(x)=x^{3}+2 x^{2}-9 x-18 $$
View solution Problem 16
For the following exercises, find the degree and leading coefficient for the given polynomial. $$ x^{2}(2 x-3)^{2} $$
View solution Problem 17
For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies jointly as \(x, z,\) and \(w\) and when \(x=1, \
View solution Problem 17
For the following exercises, find the inverse of the functions. $$ f(x)=\sqrt{3-4 x} $$
View solution