Question

\(g(x)=-x^2-1\)

Short answer

The vertex of the function \(g(x) = -x^2 - 1\) is at point (0,-1) and the axis of symmetry is the vertical line given by \(x=0\).

Step-by-step solution

Identify the Coefficients

In the given function \(g(x) = -x^2 - 1\), identify the coefficients 'a', 'b' and 'c'. Here, 'a' is -1 (the coefficient of \(x^2\)), 'b' is 0 (as there is no 'x' term), and 'c' is -1 (the constant term).

Find the vertex of the parabola

The vertex of the parabola given by a function in the form \(f(x) = ax^2 + bx + c\) is given by the point \((-b/2a , f(-b/2a))\). For the function \(g(x) = -x^2 - 1\), 'a' is -1 and 'b' is 0. So, the x-coordinate of the vertex is \(-b/2a= 0/2*(-1) = 0\). Substitute x=0 in the function to find the y-coordinate: \(g(0)= -(0)^2 - 1 = -1\). Therefore, the vertex of the parabola is (0,-1).

Find the axis of symmetry

The axis of symmetry of the parabola given by a function in the form \(f(x) = ax^2 + bx + c\) is a vertical line given by \(x=-b/2a\). For the function \(g(x) = -x^2 - 1\), 'a' is -1 and 'b' is 0. So, the equation of the axis of symmetry is \(x=0\).

Key concepts

Vertex of a Parabola

The vertex of a parabola is an important concept in understanding quadratic functions. It acts as the turning point, or the maximum or minimum of the parabola’s curve. For a function of the form \(f(x) = ax^2 + bx + c\), the vertex can be found using the formula:

• x-coordinate: \(-\frac{b}{2a}\)
• y-coordinate: substitute the x-value back into the function to get \(f(-\frac{b}{2a})\)

In our example, the function is \(g(x) = -x^2 - 1\). Here, the coefficients are \(a = -1\), \(b = 0\), and \(c = -1\).
With this information, the x-coordinate of the vertex is \(-\frac{0}{2(-1)}=0\). By plugging \(x = 0\) back into the original equation, \(g(x) = -(0)^2 - 1 = -1\), we find the y-coordinate. Thus, the vertex of this parabola is located at the point \((0, -1)\).
This gives the smallest value the function can take, since it's a downward-opening parabola.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. It divides the parabola into two symmetrical halves, making it a central concept in the study of quadratic functions. The axis of symmetry for the parabola described by the function \(f(x) = ax^2 + bx + c\) can be determined via the formula:

• \(x = -\frac{b}{2a}\)

For the function \(g(x) = -x^2 - 1\), the coefficients are \(a = -1\) and \(b = 0\). Using the formula, the axis of symmetry is given by \(x = -\frac{0}{2(-1)} = 0\).
This means the parabola is perfectly symmetric around the y-axis, as its axis of symmetry is the line \(x = 0\). Every point on the parabola has a corresponding point on the opposite side of this line, at equal horizontal distances.

Coefficients in Quadratic Equations

In quadratic equations, coefficients are the numbers in front of the variables \(x^2\), \(x\), and the constant term. These are essential in shaping the characteristics of the parabola. For any quadratic equation of the form \(f(x) = ax^2 + bx + c\):

• \(a\), the leading coefficient, affects the direction and width of the parabola. If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.
• \(b\) determines the axis of symmetry and influences the vertex’s position horizontally.
• \(c\) is the constant term, which affects the y-intercept of the parabola.

In the equation \(g(x) = -x^2 - 1\), \(a = -1\), meaning the parabola opens downward. The \(b = 0\) implies the axis of symmetry is the y-axis, and \(c = -1\) indicates the parabola intersects the y-axis at the point \((0, -1)\). Understanding these coefficients allows us to analyze and sketch the behavior of quadratic functions.