1. What is an x-intercept of the graph of a quadratic function? How many possible x-intercepts does the graph of a quadratic function have? Explain?

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**Question:**

1. What is an x-intercept of the graph of a quadratic function? How many possible x-intercepts does the graph of a quadratic function have? Explain.

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**Explanation:**

An **x-intercept** is a point where the graph of a function crosses the x-axis. For a quadratic function, which is typically in the form \( ax^2 + bx + c = 0 \), the x-intercepts are the values of \( x \) that make the equation equal to zero.

A quadratic function can have:

- **Two x-intercepts**: This occurs when the quadratic equation has two distinct real roots. The graph of the quadratic function will intersect the x-axis at two points.
  
- **One x-intercept**: This happens when the quadratic equation has exactly one real root (i.e., a "double root"). The graph will touch the x-axis at just one point, indicating the vertex of the parabola is on the x-axis.
  
- **No x-intercepts**: This is the case when there are no real roots, meaning the solutions are complex numbers. Here, the graph does not cross or touch the x-axis at any point.

These scenarios are determined by the discriminant \( b^2 - 4ac \) from the quadratic formula \(\frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\):
- If \( b^2 - 4ac > 0 \), there are two distinct real roots.
- If \( b^2 - 4ac = 0 \), there is one real root.
- If \( b^2 - 4ac < 0 \), there are no real roots.
Transcribed Image Text:**Question:** 1. What is an x-intercept of the graph of a quadratic function? How many possible x-intercepts does the graph of a quadratic function have? Explain. --- **Explanation:** An **x-intercept** is a point where the graph of a function crosses the x-axis. For a quadratic function, which is typically in the form \( ax^2 + bx + c = 0 \), the x-intercepts are the values of \( x \) that make the equation equal to zero. A quadratic function can have: - **Two x-intercepts**: This occurs when the quadratic equation has two distinct real roots. The graph of the quadratic function will intersect the x-axis at two points. - **One x-intercept**: This happens when the quadratic equation has exactly one real root (i.e., a "double root"). The graph will touch the x-axis at just one point, indicating the vertex of the parabola is on the x-axis. - **No x-intercepts**: This is the case when there are no real roots, meaning the solutions are complex numbers. Here, the graph does not cross or touch the x-axis at any point. These scenarios are determined by the discriminant \( b^2 - 4ac \) from the quadratic formula \(\frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\): - If \( b^2 - 4ac > 0 \), there are two distinct real roots. - If \( b^2 - 4ac = 0 \), there is one real root. - If \( b^2 - 4ac < 0 \), there are no real roots.
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a) What is x-intercept of the graph if a quadratic function? 

b) How many possible x -intercepts does graph of a quadratic function have? 

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