Function graphs and solution sets

Idea

A function sends an input $x$ to $f(x)$. An equation or an inequality is a condition on that graph.

The solutions are the intersections with the $x$-axis.

The solutions are the intervals where the graph lies above the $x$-axis.

The same graph can first be changed by

and then read again for zeros and for the intervals above the axis. The transform is the picture of a whole curve you can operate on. The quadratic and the polynomial show how the number of zeros, and their multiplicity, change the intersections and the sign intervals.

Interaction

• Graph transform: drag Vertical scale a, Horizontal scale b, Horizontal shift h, and Vertical shift k, and compare the ghost of $f(x)$ with the transformed curve. Switch Basis f(x) among $x$, $x^2$, $x^3$, and $|x|$
• Quadratic equation: drag Coefficient a, Coefficient b, and Coefficient c, and watch the intersections with the $x$-axis, the vertex, and $\Delta =b^2-4ac$ move together
• Polynomial inequality: drag Root r1, Root r2, and Root r3. Each of Multiplicity m1, Multiplicity m2, and Multiplicity m3 is only 1 or 2. Watch the graph cross or touch the $x$-axis, and read the gold segments on the number line, where the polynomial is positive

Suggested order: Graph transform, then Quadratic equation, then Polynomial inequality.

What to notice

• $f(x)=0$ and $f(x)>0$ are the intersections and the sign intervals. A quadratic and a polynomial differ in how many zeros they have and in the multiplicity, and you read the graph the same way.
• A transformation first moves and reshapes the whole curve. Asking how it meets the $x$-axis is another reading of that same picture, and the solution set is that reading written out.
• Crossing the $x$-axis or only touching it changes how the sign intervals are cut. The intersections after a transformation are the same language: the graph against the $x$-axis.

Related

• Rational functions and asymptotes
• A function and its derivative
• Function graph transformations
• Completing the square
• Polynomial roots and multiplicity

Further reading

• Equation (Wikipedia)
• Inequality (mathematics) (Wikipedia)