Question

use the aleks graphing calculator to find the x-intercept(s) and vertex for the quadratic function. f(x)=4x² + 6x - 1 round to the nearest hundredth if necessary. if there is more than one x-intercept, separate them with commas. if applicable, click on
one\. x-intercept(s): vertex: (-0.75, -3.25)

Explanation:

Step1: Find x-intercepts (set \( f(x)=0 \))

We have the quadratic equation \( 4x^2 + 6x - 1 = 0 \). Use the quadratic formula \( x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = 6 \), \( c=-1 \).
First, calculate the discriminant: \( \Delta = b^2 - 4ac = 6^2 - 4(4)(-1)=36 + 16 = 52 \).
Then, \( x=\frac{-6\pm\sqrt{52}}{2(4)}=\frac{-6\pm2\sqrt{13}}{8}=\frac{-3\pm\sqrt{13}}{4} \).
Calculate the two roots:

• For the plus sign: \( x=\frac{-3+\sqrt{13}}{4}\approx\frac{-3 + 3.6055}{4}\approx\frac{0.6055}{4}\approx0.15 \)
• For the minus sign: \( x=\frac{-3-\sqrt{13}}{4}\approx\frac{-3 - 3.6055}{4}\approx\frac{-6.6055}{4}\approx - 1.65 \)

Step2: Verify vertex (given as \( (-0.75, -3.25) \))

The x-coordinate of the vertex of \( ax^2+bx+c \) is \( x = -\frac{b}{2a}=-\frac{6}{2(4)}=-0.75 \).
Substitute \( x = -0.75 \) into \( f(x) \): \( f(-0.75)=4(-0.75)^2+6(-0.75)-1=4(0.5625)-4.5 - 1 = 2.25 - 4.5 - 1=-3.25 \), which matches the given vertex.

Answer:

x-intercept(s): \( -1.65, 0.15 \)
vertex: \( (-0.75, -3.25) \)