Problem 9
Question
For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $$ f(x)=x^{2}+5 x-2 $$
Step-by-Step Solution
Verified Answer
The vertex is \((-\frac{5}{2}, -\frac{33}{4})\).
1Step 1: Identify the Coefficients
In the quadratic function \( f(x) = x^2 + 5x - 2 \), identify the coefficients as follows: \( a = 1 \), \( b = 5 \), and \( c = -2 \).
2Step 2: Find the Vertex
Use the vertex formula \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex. Substitute \( a = 1 \) and \( b = 5 \). \[ x = -\frac{5}{2(1)} = -\frac{5}{2} \] Substitute \( x = -\frac{5}{2} \) into the function to find the y-coordinate: \[ f\left(-\frac{5}{2}\right) = \left(-\frac{5}{2}\right)^2 + 5\left(-\frac{5}{2}\right) - 2 \] Simplify: \[ f\left(-\frac{5}{2}\right) = \frac{25}{4} - \frac{25}{2} - 2 \] Convert to common denominator: \[ \frac{25}{4} - \frac{50}{4} - \frac{8}{4} = -\frac{33}{4} \] So the vertex is \( \left(-\frac{5}{2}, -\frac{33}{4}\right) \).
3Step 3: Rewrite in Vertex Form
The vertex form of a quadratic function is \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. Since we have \( a=1 \), \( h=-\frac{5}{2} \), and \( k = -\frac{33}{4} \), the vertex form of the function is \[ f(x) = \left(x + \frac{5}{2}\right)^2 - \frac{33}{4} \].
Key Concepts
Vertex FormStandard FormVertex of a Parabola
Vertex Form
The vertex form of a quadratic function is an essential way to express the equation, especially useful for identifying the vertex of the parabola. The general expression for the vertex form is \( f(x) = a(x-h)^2 + k \). Here, the point \((h, k)\) represents the vertex of the parabola, while \(a\) indicates the parabola's direction and width.
Using the vertex form, you can quickly determine the key features of a parabola without having to complete the square or apply the vertex formula. For instance, the parabola opens upwards if \(a > 0\) and downwards if \(a < 0\). Also, the vertex \((h, k)\) can be directly read from the equation:
Using the vertex form, you can quickly determine the key features of a parabola without having to complete the square or apply the vertex formula. For instance, the parabola opens upwards if \(a > 0\) and downwards if \(a < 0\). Also, the vertex \((h, k)\) can be directly read from the equation:
- \(h\) is the horizontal shift from the origin.
- \(k\) is the vertical shift from the origin.
Standard Form
The standard form of a quadratic function is perhaps the most recognizable and common way to express quadratic equations. A quadratic function in the standard form is written as \( f(x) = ax^2 + bx + c \). Here:
This form is straightforward to work with when performing algebraic operations, such as addition, subtraction, or factoring. When given in standard form, finding the roots or zeros of the function often involves using the quadratic formula, factoring, or graphing.
While the standard form makes some operations simpler, it doesn't readily show the vertex or easily describe the direction of the open. Thus, sometimes it needs to be converted into vertex form for better insight into the graph's characteristics.
- \(a\) is the quadratic coefficient.
- \(b\) is the linear coefficient.
- \(c\) is the constant term.
This form is straightforward to work with when performing algebraic operations, such as addition, subtraction, or factoring. When given in standard form, finding the roots or zeros of the function often involves using the quadratic formula, factoring, or graphing.
While the standard form makes some operations simpler, it doesn't readily show the vertex or easily describe the direction of the open. Thus, sometimes it needs to be converted into vertex form for better insight into the graph's characteristics.
Vertex of a Parabola
The vertex of a parabola is one of the most important points on its graph. It is the point where the parabola changes direction, serving as either the maximum or minimum value of the function, depending on its orientation.
For a quadratic function in standard form \( f(x) = ax^2 + bx + c \), the vertex can be calculated using the vertex formula \( x = -\frac{b}{2a} \). This gives the x-coordinate, and substituting this value back into the function provides the y-coordinate, forming the vertex \((x, y)\).
The vertex provides insights into the parabola's geometry:
For a quadratic function in standard form \( f(x) = ax^2 + bx + c \), the vertex can be calculated using the vertex formula \( x = -\frac{b}{2a} \). This gives the x-coordinate, and substituting this value back into the function provides the y-coordinate, forming the vertex \((x, y)\).
The vertex provides insights into the parabola's geometry:
- If \(a > 0\), the parabola opens upwards, and the vertex is the minimum point.
- If \(a < 0\), the parabola opens downwards, and the vertex is the maximum point.
Other exercises in this chapter
Problem 9
For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(2 x^{2}-3 x+2\right) \div(x+2) $$
View solution Problem 9
For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ C(t)=2 t(t-3)(t+1)^{2} $$
View solution Problem 9
Identify the function as a power function, a polynomial function, or neither. $$f(x)=\frac{x^{2}}{x^{2}-1}$$
View solution Problem 10
For the following exercises, write an equation describing the relationship of the given variables. \(y\) varies inversely as \(x\) and when \(x=4, y=2\).
View solution