Examkin

Soru

Zorluk: OrtaQuadratic Equations

In the xyxy-plane, the graph of the quadratic function f(x)=−x2+bx+cf(x) = -x^2 + bx + c, where bb and cc are constants, has its vertex at (4,9)(4, 9). What is the positive difference between the two xx-intercepts of the graph?

Cevap: 6

Cevap

The correct answer is 6.
The quadratic function can be represented in vertex form as f(x)=−(x−4)2+9f(x) = -(x - 4)^2 + 9 since the leading coefficient is −1-1 and the vertex is at (4,9)(4, 9). Setting the function equal to zero to find the xx-intercepts gives (x−4)2=9(x - 4)^2 = 9, which yields x=7x = 7 and x=1x = 1. The positive difference between these intercepts is 7−1=67 - 1 = 6.

Adım Adım Çözüm

1
Write the quadratic function in vertex form using the given vertex (4,9)(4, 9) and the leading coefficient.
f(x)=−(x−4)2+9f(x) = -(x - 4)^2 + 9
The vertex form of a quadratic function is f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex. Since the coefficient of x2x^2 is −1-1, we have a=−1a = -1, h=4h = 4, and k=9k = 9.
2
Set f(x)=0f(x) = 0 to find the xx-intercepts.
(x−4)2=9(x - 4)^2 = 9
The xx-intercepts of a graph are the points where f(x)=0f(x) = 0.
3
Solve for xx.
x=7x = 7 and x=1x = 1
Taking the square root of both sides gives x−4=±3x - 4 = \pm 3, which results in x=7x = 7 and x=1x = 1.
4
Find the positive difference between the two xx-intercepts.
6
Subtract the smaller xx-intercept from the larger xx-intercept: 7−1=67 - 1 = 6.

Anahtar Kavram

Finding the intercepts of a quadratic function using its vertex form

Alternatif Yöntem

Alternatively, expand the vertex form f(x)=−(x−4)2+9f(x) = -(x - 4)^2 + 9 to get f(x)=−(x2−8x+16)+9=−x2+8x−7f(x) = -(x^2 - 8x + 16) + 9 = -x^2 + 8x - 7. Factoring this expression gives f(x)=−(x−7)(x−1)f(x) = -(x - 7)(x - 1). The roots are x=7x = 7 and x=1x = 1, and their difference is 7−1=67 - 1 = 6.
Tahmini Süre:1m 30s
Bu soruyu puanla

Konular

Quadratic Equations

Bu konudan daha fazla soru