Respuesta :
The graph of which function has an axis of symmetry at x = -1/4 is :
f(x) = 2x² + x – 1
Further explanation
Discriminant of quadratic equation ( ax² + bx + c = 0 ) could be calculated by using :
D = b² - 4 a c
From the value of Discriminant , we know how many solutions the equation has by condition :
D < 0 → No Real Roots
D = 0 → One Real Root
D > 0 → Two Real Roots
Let us now tackle the problem!
An axis of symmetry of quadratic equation y = ax² + bx + c is :
[tex]\large {\boxed {x = \frac{-b}{2a} } }[/tex]
Option 1 :
f(x) = 2x² + x – 1 → a = 2 , b = 1 , c = -1
Axis of symmetry → [tex]x = \frac{-b}{2a} = \frac{-1}{2(2)} = -\frac{1}{4}[/tex]
Option 2 :
f(x) = 2x² – x + 1 → a = 2 , b = -1 , c = 1
Axis of symmetry → [tex]x = \frac{-b}{2a} = \frac{-(-1)}{2(2)} = \frac{1}{4}[/tex]
Option 3 :
f(x) = x² + 2x – 1 → a = 1 , b = 2 , c = -1
Axis of symmetry → [tex]x = \frac{-b}{2a} = \frac{-2}{2(1)} = -1[/tex]
Option 4 :
f(x) = x² – 2x + 1 → a = 1 , b = -2 , c = 1
Axis of symmetry → [tex]x = \frac{-b}{2a} = \frac{-(-2)}{2(1)} = 1[/tex]
Learn more
- Solving Quadratic Equations by Factoring : https://brainly.com/question/12182022
- Determine the Discriminant : https://brainly.com/question/4600943
- Formula of Quadratic Equations : https://brainly.com/question/3776858
Answer details
Grade: High School
Subject: Mathematics
Chapter: Quadratic Equations
Keywords: Quadratic , Equation , Discriminant , Real , Number
The graph of function [tex]\boxed{f(x)=2x^{2}+x-1}[/tex] has an axis of symmetry as [tex]\boxed{x=-\frac{1}{4}}[/tex].
Further explanation:
The standard form of a quadratic equation is as follows:
[tex]\boxed{f(x)=ax^{2}+bx+c}[/tex]
The vertex form of a quadratic equation is as follows:
[tex]\boxed{g(x)=a(x-h)^{2}+k}[/tex]
Axis of symmetry is the line which divides the graph of the parabola in two perfect halves.
The formula for axis of symmetry of a quadratic function is given as follows:
[tex]\boxed{x=-\dfrac{b}{2a}}[/tex]
The first function is given as follows:
[tex]f(x)=2x^{2}+x-1[/tex]
The above function is in standard form with [tex]a=2[/tex], [tex]b=1[/tex] and [tex]c=-1[/tex].
Then its axis of symmetry is calculated as,
[tex]\begin{aligned}x&=-\dfrac{b}{2a}\\&=-\dfrac{1}{2\times2}\\&=-\dfrac{1}{4}\end{aligned}[/tex]
The axis of symmetry of first function is [tex]x=-\frac{1}{4}[/tex].
Express the function [tex]f(x)=2x^{2}+x-1[/tex] in its vertex form,
[tex]\begin{aligned}f(x)&=2x^{2}+x-1\\&=(\sqrt{2}x)^{2}+\left(2\times \sqrt{2}x\times \dfrac{1}{2\sqrt{2}}\right)-1+\left(\dfrac{1}{2\sqrt{2}}\right)^{2}-\left(\dfrac{1}{\sqrt{2}}\right)^{2}\\&=\left(\sqrt{2}x+\dfrac{1}{2\sqrt{2}}\right)^{2}-1-\dfrac{1}{8}\\&=\left[\sqrt{2}\left(x+\dfrac{1}{4}\right)\right]^{2}-\dfrac{9}{8}\\&=2\left(x-\left(-\dfrac{1}{4}\right)\right)^{2}-\dfrac{9}{8}\end{aligned}[/tex]
The above equation is in the vertex form with [tex]a=2[/tex], [tex]h=-\dfrac{1}{4}[/tex] and [tex]k=-\dfrac{9}{8}[/tex].
Therefore, its axis of symmetry is given as,
[tex]\begin{aligned}x&=h\\x&=-\dfrac{1}{4}\end{aligned}[/tex]
The graph of function [tex]f(x)=2x^{2}+x-1[/tex] is shown in Figure 1.
The second function is given as follows:
[tex]f(x)=2x^{2}-x+1[/tex]
The above function is in standard form with [tex]a=2[/tex], [tex]b=-1[/tex] and [tex]c=1[/tex].
Then its axis of symmetry is calculated as,
[tex]\begin{aligned}x&=-\dfrac{b}{2a}\\&=-\dfrac{(-1)}{2\times2}\\&=\dfrac{1}{4}\end{aligned}[/tex]
The axis of symmetry of second function is [tex]x=\frac{1}{4}[/tex].
The third function is given as follows:
[tex]f(x)=x^{2}+2x-1[/tex]
The above function is in standard form with [tex]a=1[/tex], [tex]b=2[/tex] and [tex]c=-1[/tex].
Then its axis of symmetry is calculated as,
[tex]\begin{aligned}x&=-\dfrac{b}{2a}\\&=-\dfrac{2}{2\times1}\\&=-1\end{aligned}[/tex]
The axis of symmetry of third function is [tex]x=-1[/tex].
The fourth function is given as follows:
[tex]f(x)=x^{2}-2x+1[/tex]
The above function is in standard form with [tex]a=1[/tex], [tex]b=-2[/tex] and [tex]c=1[/tex].
Then its axis of symmetry is calculated as,
[tex]\begin{aligned}x&=-\dfrac{b}{2a}\\&=-\dfrac{-2}{2\times1}\\&=1\end{aligned}[/tex]
The axis of symmetry of fourth function is [tex]x=1[/tex].
Therefore, the function [tex]\boxed{f(x)=2x^{2}+x-1}[/tex] has an axis of symmetry as [tex]\boxed{x=-\frac{1}{4}}[/tex].
Learn more:
1. A problem on graph https://brainly.com/question/2491745
2. A problem on function https://brainly.com/question/9590016
3. A problem on axis of symmetry https://brainly.com/question/1286775
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Functions
Keywords:Graph, function, axis, f(x), 2x^2+x-1, axis of symmetry, symmetry, vertex, perfect halves, graph of a function, x =- 1/4.
