Understanding how function zeros and intersection points in graphs are related can be tough for 10th graders. Here are some main challenges they face:

1. Finding Zeros:

   • Many students have a hard time figuring out what the zeros (or roots) of functions are.
   • Zeros are the values of $x$ that make $f(x) = 0$.
   • Working with polynomials can feel overwhelming.

2. Reading Graphs:

   • Students often struggle to connect zeros to the graphs, where zeros appear as $x$-intercepts.
   • It’s not always easy to see that these points show where the function crosses or touches the $x$-axis.

3. Finding Intersection Points:

   • When working with more than one function, students need to find intersection points.
   • This means they have to set the functions equal to each other, like $f(x) = g(x)$.
   • Solving these equations can be complicated, especially with harder polynomials or curved functions.

Despite these challenges, students can get better at this by:

• Practicing Graphing: The more they graph functions, the easier it will be to understand zeros and what they mean on a graph.

• Using Technology: Tools like graphing calculators and software can help them see how functions change and where they intersect, making it clearer.

• Improving Algebra Skills: Getting better at algebra will help them find zeros and solve equations more easily.