Determine whether each statement is true or false. If false, explain why. The graph of y = x² + 2 has no x-intercepts.

Verified step by step guidance

1. Recall that the x-intercepts of a graph are the points where the graph crosses the x-axis. At these points, the value of y is zero, so we set the equation equal to zero to find the x-intercepts.
2. Set the equation equal to zero: \(y = x^2 + 2\) becomes \$0 = x^2 + 2$.
3. Solve for \(x\): subtract 2 from both sides to get \(x^2 = -2\).
4. Analyze the equation \(x^2 = -2\). Since \(x^2\) represents a square of a real number, it cannot be negative. Therefore, there are no real solutions for \(x\).
5. Conclude that because there are no real solutions for \(x\) when \(y=0\), the graph of \(y = x^2 + 2\) has no x-intercepts, so the statement is true.

Key Concepts

X-Intercepts of a Graph

X-intercepts are points where the graph crosses the x-axis, meaning the y-value is zero. To find x-intercepts, set y = 0 and solve for x. If no real solutions exist, the graph has no x-intercepts.

Quadratic Functions and Their Graphs

A quadratic function has the form y = ax^2 + bx + c and graphs as a parabola. The sign of 'a' determines if it opens upward or downward. The vertex and axis of symmetry help understand its shape and position.

Solving Quadratic Equations

To find roots of a quadratic equation, set it equal to zero and solve using factoring, completing the square, or the quadratic formula. The discriminant (b^2 - 4ac) indicates the number and type of solutions.