Sign pattern and graph interpretation

1. Find the x-intercepts of the graph. 2. Mark regions above and below the x-axis. 3. Read positive and negative intervals. 4. State the zeros at the intercepts.

A rough sketch should show x-intercepts, regions above and below the axis, and the corresponding sign of the polynomial.

A common wrong answer is to treat every x-intercept as a turning point. A graph may cross the axis or just touch it, so the movement matters.

The sign pattern of a polynomial tells whether the graph lies above or below the x-axis in different intervals. The x-intercepts show the zeros, and the shape of the graph helps interpret the number of real zeros.

The sign pattern of a polynomial tells whether the graph lies above or below the x-axis in different intervals. The x-intercepts show the zeros, and the shape of the graph helps interpret the number of real zeros. Above x-axis implies p(x)>0, on x-axis implies p(x)=0, and below x-axis implies p(x)<0. If a graph crosses the x-axis twice, the polynomial has two real zeros.

Graphs and signs work together in polynomial study. If the graph is above the x-axis, the polynomial values are positive there. If it is below the x-axis, the values are negative. Where the graph meets the x-axis, the value is zero. This helps students answer interpretation questions quickly without full algebraic solving. Above x-axis implies p(x)>0, on x-axis implies p(x)=0, and below x-axis implies p(x)<0. If a graph crosses the x-axis at x=1 and x=3, then the polynomial is zero at those points. Between and outside those points, the sign may change depending on the graph's direction. Interpret a graph, identify zeros, state intervals of positivity or negativity, or connect graph shape with algebraic values. A common wrong answer is to treat every x-intercept as a turning point. A graph may cross the axis or just touch it, so the movement matters.