In Exercises 17-34, write the quadratic function in standard form and sketch its graph. Identify

step 1 given the function f of x equals x squared minus x plus five fourths to get this into standard f

In Exercises 17-34, write the quadratic function in standard...

Key Concepts

x-intercepts

x-intercepts are the points where the graph of the quadratic function crosses the x-axis, which occur when the output f(x) equals zero. Calculating the x-intercepts involves solving the quadratic equation, often through methods such as factoring, completing the square, or using the quadratic formula, thereby identifying the real roots of the function.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. For a quadratic function in standard form, this line is represented by x = h, which helps in predicting the behavior of the function on either side of the vertex.

Quadratic Function

A quadratic function is a second-degree polynomial typically written as f(x) = ax² + bx + c, where a, b, and c are constants and a is non-zero. Its graph forms a parabola that either opens upward or downward depending on the sign of a, and it features characteristic elements such as a vertex, axis of symmetry, and intercepts.

Standard Form of a Quadratic Function

Rewriting a quadratic function in its standard form, f(x) = a(x - h)² + k, makes it easier to identify the vertex and analyze the graph. Here, (h, k) represents the vertex of the parabola, and the form clearly shows the transformation of the basic parabola, including its vertical shift and horizontal translation.

Vertex

The vertex is the turning point of the parabola and represents the maximum or minimum value of the quadratic function. In the standard form f(x) = a(x - h)² + k, the vertex is located at (h, k). It is crucial for pinpointing the precise location of the parabola's peak or trough when graphing the function.

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