In Exercises 35-42, use a graphing utility to graph the...

In Exercises $35-42,$ use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and $x$ -intercept(s). Then check your results algebraically by writing the quadratic function in standard form.
$$f(x)=2 x^{2}-16 x+32$$

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial function typically written as f(x) = ax² + bx + c. Its graph is a parabola, which opens upward if a > 0 and downward if a < 0. Understanding quadratic functions is fundamental in algebra because they model a wide range of phenomena and are characterized by their curvature, symmetry, and vertex.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. Its equation is given by x = -b/(2a), which coincides with the x-coordinate of the vertex. This concept is important because it highlights the inherent symmetry in quadratic functions.

X-intercepts

X-intercepts are the points where the graph of the quadratic function crosses the x-axis, meaning the output value f(x) is zero. Finding the x-intercepts typically involves solving the quadratic equation ax² + bx + c = 0 using factoring, completing the square, or the quadratic formula.

Standard Form

The standard form of a quadratic function is expressed as f(x) = ax² + bx + c. Converting a quadratic function to this form simplifies the process of analyzing its key features, such as the vertex, axis of symmetry, and intercepts, and is essential for verifying graphical interpretations with algebraic methods.

Vertex

The vertex of a quadratic function is the point where the parabola reaches its maximum or minimum value. It represents the turning point of the graph. The vertex can be found algebraically using formulas like h = -b/(2a) for the x-coordinate and evaluating f(h) for the y-coordinate, or by rewriting the function in vertex form.

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