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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경
3장, 문제 7

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

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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.
Set the equation equal to zero: \(y = x^2 + 2\) becomes \(0 = x^2 + 2\).
Solve for \(x\): subtract 2 from both sides to get \(x^2 = -2\).
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 \(x\).
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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
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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.
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Graphs of Logarithmic Functions

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.
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Solving Quadratic Equations by Factoring