Question

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range.

$f (x) = - (x + 1)^{2} + 4$

Accepted Answer

Rewrite the given quadratic function in standard vertex form: $$ f(x) = (x + 4)^2 - 2 . $$Here, the vertex form is $$ f(x) = a(x - h)^2 + k $$, where $$ (h, k)  is $$the vertex. From the equation, the vertex is $$ (-4, -2) . $$Identify the axis of symmetry. The axis of symmetry is a vertical line that passes through the x-coordinate of the vertex. Therefore, the equation for the axis of symmetry is $$ x = -4 . $$Find the x-intercepts by setting $$ f(x) = 0 $$ and solving for $$ x $$: $$ 0 = (x + 4)^2 - 2 . $$Add 2 to both sides to get $$ (x + 4)^2 = 2 . $$Take the square root of both sides, remembering to include both the positive and negative roots, to get $$ x + 4 = \pm \sqrt{2} . $$Finally, solve for $$ x  by $$subtracting 4 from both sides: $$ x = -4 \pm \sqrt{2} . $$Find the y-intercept by setting $$ x = 0  in $$the function: $$ f(0) = (0 + 4)^2 - 2 . $$Simplify to find the y-intercept as a point $$ (0, y) . $$Determine the domain and range of the function. Since this is a quadratic function, the domain is all real numbers, $$ (-\infty, \infty) . $$The range is determined by the vertex and the direction of the parabola. Since the parabola opens upwards (the coefficient of $$ (x + 4)^2  is $$positive), the range is $$ [-2, \infty) $$, starting at the y-coordinate of the vertex.

In Exercises 1–4, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x) = (x + 4)^2 - 2

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Key Concepts

Vertex of a Quadratic Function

Axis of Symmetry

Domain and Range of Quadratic Functions