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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 7
Chapter 3, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Related Practice
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Without actually graphing, identify the type of graph that each equation has.

x2+y2=144x^2+y^2=144

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Determine whether each statement is true or false. If false, explain why. The midpoint of the segment joining (0, 0) and (4, 4) is 2.