Question

1. what are the slope and the y-intercept of the graph of the linear function shown on the grid?

slope: _____________ y-intercept: ______________________

Explanation:

Step1: Calculate the slope

The formula for slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). We have two points \((-6, 0)\) and \((2, -2)\). Let \((x_1, y_1)=(-6, 0)\) and \((x_2, y_2)=(2, -2)\). Then \( m=\frac{-2 - 0}{2 - (-6)}=\frac{-2}{8}=-\frac{1}{4} \).

Step2: Find the y - intercept

The equation of a line is \( y = mx + b \), where \( b \) is the y - intercept. We can use one of the points, say \((-6, 0)\), and the slope \( m =-\frac{1}{4}\) to find \( b \). Substitute into the equation: \( 0=-\frac{1}{4}(-6)+b \). Simplify \( 0=\frac{6}{4}+b \), then \( b =-\frac{6}{4}=-\frac{3}{2} \)? Wait, no, let's use the other point \((2, -2)\). Substitute \( x = 2 \), \( y=-2 \), and \( m =-\frac{1}{4}\) into \( y=mx + b \): \(-2=-\frac{1}{4}(2)+b\). Simplify: \(-2 =-\frac{1}{2}+b\). Then \( b=-2+\frac{1}{2}=-\frac{3}{2}\)? Wait, maybe I made a mistake. Wait, let's check the graph again. Wait, when \( x = 0 \), what's \( y \)? Let's use the two - point formula correctly. Wait, the two points are \((-6,0)\) and \((2, - 2)\). The slope is \( m=\frac{-2 - 0}{2-(-6)}=\frac{-2}{8}=-\frac{1}{4}\). Now, using the point - slope form \( y - y_1=m(x - x_1) \) with \((x_1,y_1)=(-6,0)\): \( y-0 =-\frac{1}{4}(x + 6) \), \( y=-\frac{1}{4}x-\frac{6}{4}=-\frac{1}{4}x-\frac{3}{2} \). Wait, but when \( x = 0 \), \( y=-\frac{3}{2}=-1.5 \). Wait, but maybe I misread the graph. Wait, let's check the coordinates again. Wait, the point \((2,-2)\) and \((-6,0)\). Wait, maybe the y - intercept is at \( (0, - 1.5) \) or \( (0,-\frac{3}{2}) \). But wait, let's recalculate the slope again. Wait, the change in y: from \((-6,0)\) to \((2,-2)\), \( \Delta y=-2 - 0=-2 \), \( \Delta x=2-(-6)=8 \), so slope is \( \frac{-2}{8}=-\frac{1}{4} \). Then, to find the y - intercept, we can use the equation \( y=mx + b \). Let's plug in \( x = 2 \), \( y=-2 \), \( m =-\frac{1}{4} \): \(-2=-\frac{1}{4}(2)+b\), \(-2=-\frac{1}{2}+b\), so \( b=-2+\frac{1}{2}=-\frac{3}{2}=-1.5 \).

Wait, but maybe I made a mistake in the point selection. Wait, looking at the graph, the line passes through \((-6,0)\) and \((2, - 2)\). Let's calculate the slope again: \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-2-0}{2 - (-6)}=\frac{-2}{8}=-\frac{1}{4}\). Now, for the y - intercept, when \( x = 0 \), we can also calculate it by looking at the graph. Wait, maybe the correct y - intercept is \( - \frac{3}{2}\) or let's check with another approach. Wait, the slope is \( -\frac{1}{4} \), and using the point \((-6,0)\), the equation is \( y=-\frac{1}{4}(x + 6) \). When \( x = 0 \), \( y=-\frac{6}{4}=-\frac{3}{2}=-1.5 \). So the slope is \( -\frac{1}{4} \) and the y - intercept is \( -\frac{3}{2} \) (or - 1.5).

Answer:

Slope: \(-\frac{1}{4}\)
y - intercept: \(-\frac{3}{2}\) (or - 1.5)