To graph the function \( y = x^2 - bx + 5 \), we can explore the effect of the parameter \( b \) on the graph's shape and position.

1. **General Shape**: The function \( y = x^2 - bx + 5 \) is a quadratic equation, so its graph will be a parabola opening upwards, as the coefficient of \( x^2 \) is positive.

2. **Vertex**: The vertex of a parabola given by \( y = ax^2 + bx + c \) is located at \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( c = 5 \), so:
   \[
   x = -\frac{-b}{2 \cdot 1} = \frac{b}{2}.
   \]
   Therefore, the x-coordinate of the vertex depends on the value of \( b \), affecting the horizontal location of the parabola’s vertex.

3. **Intercepts**:
   - **Y-Intercept**: At \( x = 0 \), \( y = 5 \), so the y-intercept is always \( (0, 5) \).
   - **X-Intercepts**: To find x-intercepts, we solve \( x^2 - bx + 5 = 0 \). The solutions (roots) will depend on \( b \) and are given by the quadratic formula:
     \[
     x = \frac{b \pm \sqrt{b^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{b \pm \sqrt{b^2 - 20}}{2}.
     \]
     The discriminant \( b^2 - 20 \) will determine the number of real roots:
     - If \( b^2 > 20 \), there are two distinct real roots.
     - If \( b^2 = 20 \), there is one real root (the vertex touches the x-axis).
     - If \( b^2 < 20 \), there are no real x-intercepts.

4. **Effect of Changing \( b \)**: Changing \( b \) shifts the parabola horizontally and affects the vertex and the roots (if they exist). Higher values of \( |b| \) shift the vertex further left or right.

Would you like me to plot the graph for a specific value of \( b \)? 

---

Here are some additional questions for practice:

1. What happens to the graph if \( b = 0 \)?
2. How does changing the value of \( c \) in a quadratic function \( y = x^2 - bx + c \) affect its graph?
3. For which values of \( b \) will the function have no real x-intercepts?
4. How would the graph change if the equation were \( y = -x^2 - bx + 5 \)?
5. Can you find the coordinates of the vertex for \( b = 4 \)?

**Tip**: Remember that the discriminant in a quadratic equation, \( b^2 - 4ac \), helps determine the nature and number of roots.

Explore how to graph the quadratic function y = x^2 - bx + 5, examining vertex position, intercepts, and the effect of parameter b on the parabola. This guide covers key algebraic concepts, the vertex formula, and the discriminant for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation