Determine whether each statement is true or false. If false, explain why. The graph of y = x^2 + 2 has no x-intercepts.

Verified step by step guidance

1. <insert step 1> Start by understanding that an x-intercept occurs where the graph of the equation crosses the x-axis, which means y = 0 at that point.>
2. <insert step 2> Set the equation y = x^2 + 2 equal to 0 to find the x-intercepts: x^2 + 2 = 0.>
3. <insert step 3> Solve the equation x^2 + 2 = 0 for x. Subtract 2 from both sides to get x^2 = -2.>
4. <insert step 4> Consider the nature of the equation x^2 = -2. Since x^2 represents a square of a real number, it cannot be negative.>
5. <insert step 5> Conclude that there are no real solutions for x, meaning the graph of y = x^2 + 2 does not cross the x-axis, confirming the statement is true.>

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form y = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the general shape and properties of parabolas is essential for analyzing their intercepts.

X-Intercepts

X-intercepts are the points where a graph crosses the x-axis, which occur when y = 0. To find x-intercepts for a quadratic function, one must solve the equation ax^2 + bx + c = 0. The nature and number of x-intercepts can be determined using the discriminant (b^2 - 4ac) from the quadratic formula.

Discriminant

The discriminant is a component of the quadratic formula that helps determine the nature of the roots of a quadratic equation. It is calculated as b^2 - 4ac. If the discriminant is positive, there are two distinct real roots (x-intercepts); if it is zero, there is one real root (a repeated x-intercept); and if it is negative, there are no real roots, indicating the graph does not intersect the x-axis.