Find a formula for the given linear function.
The graph of h intersects the graph of y = x^2 at x = -2 and x = 4
Intersections with y=x^2 at x=-2 and x=4 describes two points:
(-2, y=(-2)^2) and (4, y=4^2)
Find the line between those two points.
For x = - 2
y = ( - 2 ) ^ 2 = 4
( x1 = - 2 , y1 = 4 )
For x = 4
y = 4 ^ 2 = 16
( x2 = 4 , y2 = 16 )
The line through two distinct points (x1, y1) and (x2, y2) is given by
y = y1 + [ ( y2 - y1 ) / ( x2 - x1 ) ] * ( x - x1 )
In your case :
x1 = - 2
y1 = 4
x2 = 4
y2 = 16
y = y1 + [ ( y2 - y1 ) / ( x2 - x1 ) ] * ( x - x1 )
y = 4 + [ ( 16 - 4 ) / ( 4 - ( - 2 ) ] * x - ( - 2 ) ]
y = 4 + [ 12 / ( 4 + 2 ) ] * ( x + 2 )
y = 4 + ( 12 / 6 ) * ( x + 2 )
y = 4 + 2 * ( x + 2 )
y = 4 + 2 x + 4
y = 2 x + 8
Proof :
For x = - 2
y = 2 * ( - 2 ) + 8 = - 4 + 8 = 4
( x1 = - 2 , y1 = 4 )
For x = 4
y = 2 * 4 + 8 = 8 + 8 = 16
( x2 = 4 , y2 = 16 )
Sure! Let me put on my thinking cap... or should I say, my clown wig? *wink*
To find the formula for the linear function, let's start by finding the points of intersection between h and y = x^2. We know that h intersects y = x^2 at x = -2 and x = 4.
So, the coordinates of the points of intersection are (-2, h(-2)) and (4, h(4)). These points lie on the graph of the linear function.
Now let's find the equation of the line passing through these two points using the famous slope-intercept form: y = mx + b.
The slope (m) can be determined by finding the difference in y-coordinates divided by the difference in x-coordinates. In other words,
m = (h(4) - h(-2)) / (4 - (-2)).
The denominator simplifies to 6, so the slope becomes (h(4) - h(-2)) / 6.
As for the y-intercept (b), we can plug in one of the intersection points into the equation y = mx + b and solve for b.
Let's choose the point (4, h(4)):
h(4) = m*4 + b
b = h(4) - 4m.
Now, combining the slope formula and the y-intercept formula, we have:
y = ((h(4) - h(-2)) / 6)x + (h(4) - 4((h(4) - h(-2)) / 6)).
Simplified, this gives us:
y = ((h(4) - h(-2)) / 6)x + (h(4) - 2/3(h(4) - h(-2))).
So, the formula for the linear function h is:
y = ((h(4) - h(-2)) / 6)x + (h(4) - 2/3(h(4) - h(-2))).
Now, my friend, you have a formula that represents the given linear function h. I hope it brings some laughter to your mathematical adventures! 🤡
To find the formula for the linear function, let's first find the y-values of the points of intersection.
For x = -2:
y = (-2)^2 = 4
For x = 4:
y = (4)^2 = 16
So, the two points of intersection are (-2, 4) and (4, 16).
Now, let's find the slope of the linear function using the two points. The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
m = (16 - 4) / (4 - (-2))
m = 12 / 6
m = 2
Now that we know the slope, we can use the point-slope formula to find the equation of the linear function. The point-slope formula is given by:
y - y1 = m(x - x1)
Using the point (-2, 4):
y - 4 = 2(x - (-2))
y - 4 = 2(x + 2)
y - 4 = 2x + 4
y = 2x + 8
Therefore, the formula for the linear function is y = 2x + 8.
To find a formula for the given linear function, we need to determine the equation for the line. Since we know that the graph of h intersects the graph of y = x^2 at x = -2 and x = 4, we can find the corresponding y-values for these x-values.
At x = -2, we substitute -2 into the equation y = x^2:
y = (-2)^2
y = 4
Similarly, at x = 4, we substitute 4 into the equation y = x^2:
y = 4^2
y = 16
Now we have two points on the line, (-2, 4) and (4, 16).
To find the equation of the line passing through these two points, we can use the point-slope form of a linear equation. The point-slope form is given by: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line.
Let's calculate the slope first:
slope (m) = (y2 - y1) / (x2 - x1) [Using the points (-2, 4) and (4, 16)]
= (16 - 4) / (4 - (-2))
= 12 / 6
= 2
Now, we can use the point-slope form with either of the given points. Let's use (-2, 4):
y - 4 = 2(x - (-2))
y - 4 = 2(x + 2)
y - 4 = 2x + 4
y = 2x + 8
Therefore, the formula for the given linear function is y = 2x + 8.