Answer:
Equation of the parabola: [tex]y = 2x^{2} + 4x - 3[/tex].
Axis of symmetry: [tex]x= -1[/tex].
Coordinates of the vertex: [tex]\displaystyle \left(-1, -5\right)[/tex].
Step-by-step explanation:
The axis of symmetry and the coordinates of the vertex of a parabola can be read directly from its equation in vertex form.
[tex]y = a(x-h)^{2} +k[/tex].
The vertex of this parabola will be at [tex](h, k)[/tex]. The axis of symmetry will be [tex]x = h[/tex].
The equation in this question is in standard form. It will take some extra steps to find the vertex form of this equation before its vertex and axis of symmetry can be found. To find the vertex form, find its coefficients [tex]a[/tex], [tex]h[/tex], and[tex]k[/tex].
Expand the square in the vertex form using the binomial theorem.
[tex]y = a(x-h)^{2} +k[/tex].
[tex]y = a(x^{2} - 2hx + h^{2}) +k[/tex].
By the distributive property of multiplication,
[tex]y = (ax^{2} - 2ahx + ah^{2}) +k[/tex].
Collect the constant terms:
[tex]y = ax^{2} - 2ahx + (ah^{2} +k)[/tex].
The coefficients in front of powers of [tex]x[/tex] shall be the same in the two forms. For example, the coefficient of [tex]x^{2}[/tex] in the given equation is [tex]2[/tex]. The coefficient of [tex]x^{2}[/tex] in the equation [tex]y = ax^{2} - 2ahx + (ah^{2} +k)[/tex] is [tex]a[/tex]. The two coefficients need to be equal for the two equations to be equivalent. As a result, [tex]a = 2[/tex].
Similarly, for the term [tex]x[/tex]:
[tex]-2ah = 4[/tex].
[tex]\displaystyle h = -\frac{2}{a} = -1[/tex].
So is the case for the constant term:
[tex]ah^{2} + k = -3[/tex].
[tex]k = -3 - ah^{2} = -5[/tex].
The vertex form of this parabola will thus be:
[tex]y = 2(x -(-1))^{2} + (-5)[/tex].
The vertex of this parabola is at [tex](-1, -5)[/tex].
The axis of symmetry of a parabola is a vertical line that goes through its vertex. For this parabola, the axis of symmetry is the line [tex]x = -1[/tex].
