Question 182·Medium·Nonlinear Functions
Which choice could be the equation of the graph shown in the -plane?
For a polynomial shown in factored-form choices, read the x-intercepts from the graph to determine the linear factors, then use whether the graph crosses or only touches at each intercept to decide which factor must be squared. Finally, use the end behavior (both ends up or both ends down) to choose the correct sign of the leading coefficient.
Hints
Match intercepts to factors
Each x-intercept corresponds to a factor in a factored polynomial equation.
Look for a tangent x-intercept
If the curve touches the x-axis and turns around at an intercept, the corresponding factor has an even exponent, like .
Check the ends of the graph
Decide whether the graph rises or falls as becomes very large in the positive and negative directions to determine the sign of the leading coefficient.
Desmos Guide
Graph the curve and check intercepts
Enter each answer choice as its own equation (for example, type the option exactly as written). For each graph, identify the x-intercepts (where the curve crosses or touches the x-axis).
Check which intercept is a “touch”
Zoom near and see whether the graph crosses the x-axis or just touches it and turns around. Select the equation that touches at .
Confirm end behavior
Look far to the left and far to the right (or zoom out). Choose the equation whose graph rises on both ends, matching the given graph.
Step-by-step Explanation
Use the x-intercepts
From the graph, the x-intercepts are at , , and , so the equation must include factors , , and .
Use the “touching” behavior
At , the curve touches the x-axis and turns around (it does not cross). That means is a zero with even multiplicity, so the factor must be .
Use end behavior to choose the sign
The graph rises on both the far left and far right, which matches an even-degree polynomial with a positive leading coefficient. The only choice with zeros at , (double), and , and a positive leading coefficient is: