A **quadratic function** is a polynomial function of degree 2. A quadratic function can be written in the form f\left(x\right)=ax^{2}+bx+c where a, b, and c are real numbers.

From the graph of a quadratic function, called a **parabola**, we can identify key features including domain and range, x- and y-intercepts, and if it has a maximum or a minimum. The parabola also has the following two features that help us identify it, and that we can use when drawing the graph:

We can determine the key features of a quadratic function from its graph:

We can identify the x-intercepts of some quadratic equations by drawing the graph of the corresponding function.

The x-intercepts of a quadratic function can also be seen in a table of values, provided the right values of x are chosen and the equation has at least one real x-intercept.

Consider the quadratic function: f\left(x\right)=x^{2}-2x+1

a

Graph the function.

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b

State the axis of symmetry.

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Consider the graph of the quadratic function g\left(x\right):

a

Find the x-intercepts and y-intercept.

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b

Determine the domain and range.

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c

Describe what happens to the graph as x gets very large and positive.

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The graph shows the height, y (in feet), of a softball above ground x seconds after it was thrown in the air.

a

Find the y-intercept and describe what it means in context.

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b

Find the value of the x-intercept and describe what it means in context.

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c

Find the value of the vertex and describe what it means in context.

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d

State the domain and describe what it means in context.

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Idea summary

From the graph of a quadratic function, we can identify key features including:

Domain and range

x- and y-intercepts

Maximum or minimum function value

Vertex

Axis of symmetry