Hi! Can someone help me with these? Thanks!
1.) Determine the slope and the y-intercept of the equation of the line passing through the points (-5, 3) and (1, -3). Write the equation of the line in slope intercept form.
2.) Determine the x- and y- intercepts and use them to graph the following equation. y=x^2-2x-3
(1)
The slope is -6/6 = -1
So, in point-slope, we have
y-3 = -1(x+5)
Now massage that into slope-intercept form.
Clearly, when x=0, y = -3
Since y = (x-3)(x+1), y=0 when x is -1 or 3.
So, draw a parabola with those roots, which opens upward. Recall that the vertex occurs midway between the roots, where x = 1.
slope = (-3 - 3)/ (1+5) = -6/6 = -1
y = -x + b
3 = -1(-5) + b
b = -2
y = -x - 2
===============================
x axis(0, -3)
y axis where y = 0
x^2 - 2 x - 3 = 0
(x-3)(x+1) = 0
x = 3 or x = 1
so
axis of symmetry and vertex at x = (1+3)/2 = 2
then y vertex = 4-4 - 3 = -3
so vertex at (2 , -3) opens up
through (1,0) and (3,0)
dumb error
Of course! I can help you with both of these questions.
1.) To find the slope (m) and the y-intercept (b) of the line passing through two points, you can use the slope-intercept form of a linear equation, which is y = mx + b.
Step 1: Find the slope (m)
The formula to find the slope (m) is given by:
m = (y2 - y1) / (x2 - x1)
Using (-5, 3) and (1, -3):
m = (-3 - 3) / (1 - (-5))
m = (-6) / (1 + 5)
m = -6 / 6
m = -1
So, the slope of the line passing through (-5, 3) and (1, -3) is -1.
Step 2: Find the y-intercept (b)
Now that we have the slope, we can substitute one of the given points into the slope-intercept form (y = mx + b) to solve for the y-intercept (b).
Using the point (-5, 3):
3 = (-1)(-5) + b
3 = 5 + b
b = 3 - 5
b = -2
So, the y-intercept of the line passing through (-5, 3) and (1, -3) is -2.
Step 3: Write the equation in slope-intercept form
We now have the slope (m = -1) and the y-intercept (b = -2), so we can write the equation in slope-intercept form:
y = -1x - 2
2.) To find the x- and y-intercepts of the equation y = x^2 - 2x - 3, you can set y = 0 to find the x-intercepts and set x = 0 to find the y-intercept.
Step 1: Find the x-intercepts
Set y = 0:
0 = x^2 - 2x - 3
To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, factoring will work:
0 = (x - 3)(x + 1)
Setting each factor equal to 0 gives us two possible x-intercepts:
x - 3 = 0 --> x = 3
x + 1 = 0 --> x = -1
So the x-intercepts are x = 3 and x = -1.
Step 2: Find the y-intercept
To find the y-intercept, we set x = 0:
y = (0)^2 - 2(0) - 3
y = -3
So the y-intercept is y = -3.
Now, we can use these intercepts to graph the equation. Put the x-intercepts (-1 and 3) on the x-axis and the y-intercept (-3) on the y-axis. Then sketch the curve of the quadratic equation through these points.
I hope this helps! Let me know if you have any further questions.