Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 5)² - 4

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Identify the given quadratic function: \(f(x) = (x - 5)^2 - 4\). This is in vertex form, which is \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.

Find the vertex by comparing the given function to the vertex form. Here, \(h = 5\) and \(k = -4\), so the vertex is at the point \((5, -4)\).

Determine the axis of symmetry, which is the vertical line that passes through the vertex. The axis of symmetry is \(x = h\), so here it is \(x = 5\).

State the domain of the function. Since this is a quadratic function, the domain is all real numbers, which can be written as \((-\infty, \infty)\).

Find the range of the function. Because the parabola opens upwards (the coefficient of the squared term is positive), the range starts at the vertex's \(y\)-value and goes to infinity. So, the range is \([-4, \infty)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the graph of a quadratic function, given in vertex form as f(x) = a(x - h)^2 + k, where (h, k) is the vertex. It represents the point where the parabola changes direction.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. For a quadratic in vertex form, the axis of symmetry is x = h, where h is the x-coordinate of the vertex.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex and the direction of the parabola (upward if a > 0, downward if a < 0), starting from the vertex's y-value and extending infinitely in one direction.

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = (x - 5)2 - 4