Between which two ordered pairs does the graph of [tex]f(x)=\frac{1}{2} x^2+x-9[/tex] cross the negative [tex]x[/tex]-axis?
Quadratic formula: [tex]x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}[/tex]
A. ( [tex]-6,0[/tex] ) and ( [tex]-5,0[/tex] )
B. ( [tex]-4,0[/tex] ) and ( [tex]-3,0[/tex] )
C. ( [tex]-3,0[/tex] ) and ( [tex]-2,0[/tex] )
D. ( [tex]-2,0[/tex] ) and ( [tex]-1,0[/tex] )
Answer (1)
Apply the quadratic formula to find the roots of the equation 2 1 x 2 + x − 9 = 0 .
Identify the negative root as x = − 1 − 19 ≈ − 5.3589 .
Determine that the negative root lies between − 6 and − 5 .
Conclude that the graph crosses the negative x-axis between ( − 6 , 0 ) and ( − 5 , 0 ) , so the answer is ( − 6 , 0 ) and ( − 5 , 0 ) .
Explanation
Problem Analysis We are given the quadratic function f ( x ) = 2 1 x 2 + x − 9 and asked to find between which two ordered pairs the graph crosses the negative x -axis. This means we need to find the negative root of the equation f ( x ) = 0 and determine the two consecutive integers between which this root lies.
Applying the Quadratic Formula To find the roots of the quadratic equation 2 1 x 2 + x − 9 = 0 , we can use the quadratic formula: x = 2 a − b ± b 2 − 4 a c . In this case, a = 2 1 , b = 1 , and c = − 9 .
Calculating the Roots Plugging the values of a , b , and c into the quadratic formula, we get: x = 2 ( 2 1 ) − 1 ± 1 2 − 4 ( 2 1 ) ( − 9 ) = 1 − 1 ± 1 + 18 = − 1 ± 19 So the two roots are x 1 = − 1 − 19 and x 2 = − 1 + 19 .
Estimating the Negative Root We are interested in the negative root, which is x 1 = − 1 − 19 . Since 16 = 4 and 25 = 5 , we know that 4 < 19 < 5 . Therefore, − 1 − 5 < − 1 − 19 < − 1 − 4 , which simplifies to − 6 < − 1 − 19 < − 5 .
Determining the Interval Using a calculator, we find that 19 ≈ 4.3589 , so x 1 = − 1 − 19 ≈ − 1 − 4.3589 = − 5.3589 . This confirms that the negative root lies between − 6 and − 5 . Therefore, the graph of f ( x ) crosses the negative x -axis between the ordered pairs ( − 6 , 0 ) and ( − 5 , 0 ) .
Final Answer The graph of f ( x ) crosses the negative x -axis between the ordered pairs ( − 6 , 0 ) and ( − 5 , 0 ) .
Examples
Understanding where a function crosses the x-axis is crucial in many real-world applications. For instance, in physics, the roots of a projectile's height function can tell us when the projectile hits the ground. Similarly, in economics, the roots of a profit function can indicate break-even points. By finding the x-intercepts, we gain valuable insights into the behavior and critical points of the function, enabling informed decisions and predictions in various fields.
