*y*=2*x*2−3*x*+5. what are the x intercepts and y intercept and axis of symmetry 

Let's analyze the equation \( y = 2x^2 - 3x + 5 \) to find the x-intercepts, y-intercept, and axis of symmetry.

### 1. **X-Intercepts:**

To find the x-intercepts, set \( y = 0 \) and solve for \( x \):

\[
0 = 2x^2 - 3x + 5
\]

This is a quadratic equation. The solutions for \( x \) can be found using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For the equation \( 2x^2 - 3x + 5 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -3 \)
- \( c = 5 \)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(5)}}{2(2)}
\]

\[
x = \frac{3 \pm \sqrt{9 - 40}}{4}
\]

\[
x = \frac{3 \pm \sqrt{-31}}{4}
\]

Since the discriminant \( \sqrt{-31} \) is negative, there are no real solutions for \( x \). This means the parabola does not intersect the x-axis, so there are no x-intercepts.

### 2. **Y-Intercept:**

The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the equation:

\[
y = 2(0)^2 - 3(0) + 5 = 5
\]

So, the y-intercept is \( (0, 5) \).

### 3. **Axis of Symmetry:**

The axis of symmetry for a parabola given by \( y = ax^2 + bx + c \) is found using the formula:

\[
x = -\frac{b}{2a}
\]

Substitute \( a = 2 \) and \( b = -3 \):

\[
x = -\frac{-3}{2(2)} = \frac{3}{4} = 0.75
\]

So, the axis of symmetry is the vertical line \( x = 0.75 \).

### Summary:
- **X-Intercepts:** None (no real solutions).
- **Y-Intercept:** \( (0, 5) \).
- **Axis of Symmetry:** \( x = 0.75 \).

Would you like further clarification or have any additional questions? Here are five related questions to consider:

1. What is the vertex of the parabola \( y = 2x^2 - 3x + 5 \)?
2. How does the discriminant of a quadratic equation determine the number of x-intercepts?
3. Can you calculate the minimum or maximum value of \( y \) for this parabola?
4. How would the axis of symmetry change if the coefficient \( b \) were different?
5. What is the significance of the y-intercept in the context of real-world applications?

**Tip:** The discriminant of a quadratic equation can quickly tell you if a parabola has real, repeated, or no x-intercepts.

Discover how to find the x-intercepts, y-intercept, and axis of symmetry of the quadratic equation 2x^2 - 3x + 5. Learn the quadratic formula and its application in solving for roots. Suitable for high school students studying quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation