Question

use the aleks graphing calculator to find the vertex and x-intercept(s) for the quadratic function, f(x)=3x² -9x +8 round to the nearest hundredth if necessary. if there is more than one x-intercept, separate them with commas. if applicable, click on
one\. vertex: x-intercept(s):

Explanation:

Step1: Find the x-coordinate of the vertex

For a quadratic function \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Here, \( a = 3 \) and \( b = -9 \), so \( x = -\frac{-9}{2\times3} = \frac{9}{6} = 1.5 \).

Step2: Find the y-coordinate of the vertex

Substitute \( x = 1.5 \) into the function \( f(x) = 3x^2 - 9x + 8 \). So \( f(1.5) = 3\times(1.5)^2 - 9\times1.5 + 8 = 3\times2.25 - 13.5 + 8 = 6.75 - 13.5 + 8 = 1.25 \). Thus, the vertex is \( (1.5, 1.25) \).

Step3: Find the x-intercepts

Set \( f(x) = 0 \), so \( 3x^2 - 9x + 8 = 0 \). Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 3 \), \( b = -9 \), \( c = 8 \). The discriminant \( D = b^2 - 4ac = (-9)^2 - 4\times3\times8 = 81 - 96 = -15 \). Since the discriminant is negative, there are no real x-intercepts.

Answer:

vertex: \( (1.5, 1.25) \)
x-intercept(s): None