Transformations of Quadratic Functions

Front

What is a quadratic function?

Back

A quadratic function is a polynomial function of degree 2, typically in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

Front

What does the graph of a quadratic function look like?

Back

The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient 'a'.

Front

What is the vertex of a quadratic function?

Back

The vertex is the highest or lowest point of the parabola, depending on its orientation. It can be found using the formula (-b/2a, f(-b/2a)).

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, given by the equation x = -b/2a.

Front

What is the effect of changing the value of 'a' in a quadratic function?

Back

Changing 'a' affects the width and direction of the parabola. If |a| > 1, the parabola is narrower; if 0 < |a| < 1, it is wider. If 'a' is negative, the parabola opens downwards.

Front

What is a vertical shift in the context of quadratic functions?

Back

A vertical shift moves the graph up or down. For example, f(x) = x² + k shifts the graph up by k units if k > 0 and down by k units if k < 0.

Front

What is a horizontal shift in the context of quadratic functions?

Back

A horizontal shift moves the graph left or right. For example, f(x) = (x - h)² shifts the graph right by h units if h > 0 and left by h units if h < 0.

Front

What is a vertical compression in quadratic functions?

Back

A vertical compression occurs when the value of 'a' is between 0 and 1, making the parabola wider. For example, f(x) = (3/5)x² compresses the graph vertically.

Front

What is a reflection in the context of quadratic functions?

Back

A reflection occurs when 'a' is negative, causing the parabola to open downwards. For example, f(x) = -x² reflects the graph over the x-axis.

Front

How do you determine the vertical shift from the equation f(x) = -(x + 3)² - 5?

Back

The vertical shift is down 5 units, as indicated by the -5 in the equation.