58.4 Quadratic Transformation Effects

Quadratic Transformation Effects describes how each individual modification applied to the parent quadratic graph changes its appearance, isolating each transformation — shifting, reflecting, stretching, or compressing — as a distinct effect before considering how several such effects combine within a single expression.

Rightward Quadratic Shift

The Effect

Subtracting a positive constant from the variable before squaring moves the parent graph to the right by an amount equal to that constant, while leaving its shape and opening direction unchanged.

Why Subtraction Produces a Rightward Move

The new curve reaches the same output value that the parent curve reached at input x only when the new input is h units larger, since h must first be subtracted back down to x before squaring occurs. This delay in reaching each output value is what shifts the entire curve to the right.

Leftward Quadratic Shift

The Effect

Adding a positive constant to the variable before squaring moves the parent graph to the left by an amount equal to that constant.

Consistency with the Subtraction Convention

Because vertex form is written with subtraction inside the group, an addition sign inside the parentheses is understood as subtracting a negative value of h, keeping this leftward shift consistent with the same underlying formula used for rightward shifts.

Upward Quadratic Shift

The Effect

Adding a positive constant outside the squared group moves the parent graph upward by an amount equal to that constant, without affecting its horizontal position or shape.

Direct Effect on Output Values

Every output value produced by the parent function at a given input is increased by exactly k, which raises the entire curve vertically without changing which input produces which relative shape.

Downward Quadratic Shift

The Effect

Subtracting a positive constant outside the squared group moves the parent graph downward by an amount equal to that constant.

Direct Effect on Output Values

Every output value produced by the parent function at a given input is decreased by exactly k, which lowers the entire curve vertically.

Quadratic X-Axis Reflection

The Effect

Multiplying the entire squared expression by negative one flips the parent graph over the horizontal axis, turning an upward-opening curve into a downward-opening curve.

Every Output Value Negated

Reflection negates every output value the parent function would have produced, converting each previously positive or zero output into its negative counterpart while leaving the input values themselves unchanged.

Quadratic Vertical Stretch

The Effect

Multiplying the squared expression by a coefficient with magnitude greater than one narrows the parabola compared to the parent graph, making it rise more steeply as the input moves away from the vertex.

Effect on Output Magnitude

Each output value the parent function would have produced is scaled up in magnitude by the factor a, which makes the curve climb faster and appear visually narrower.

Quadratic Vertical Compression

The Effect

Multiplying the squared expression by a coefficient with magnitude between zero and one widens the parabola compared to the parent graph, making it rise more gradually as the input moves away from the vertex.

Effect on Output Magnitude

Each output value the parent function would have produced is scaled down in magnitude by the factor a, which makes the curve climb more slowly and appear visually wider.

Combined Quadratic Transformation

Applying Multiple Effects Together

When a quadratic expression contains a coefficient outside the squared group along with values inside and outside that group, all of the effects described above apply simultaneously: the sign and magnitude of the coefficient control reflection and stretching or compression, the value inside the group controls horizontal shift, and the value added or subtracted outside the group controls vertical shift.

Independence of the Combined Effects

Each of these effects acts independently of the others: changing the horizontal shift value does not alter the width or the vertical position, and changing the coefficient does not alter the horizontal or vertical position. This independence allows every transformed graph to be described completely by identifying each value separately.