Math Problem Statement

what’s the equation in slope intercept form

Solution

To find the equation of the tangent line in slope-intercept form, we need two pieces of information: the slope of the tangent line and the coordinates of the point where the tangent line touches the curve.

We already have the slope of the tangent line at x=1x = 1, which is 8-8.

Next, we need the coordinates of the point on the curve where x=1x = 1. We can find the yy-coordinate by substituting x=1x = 1 into the original function:

y=(1)44(1)3+2y = (1)^4 - 4(1)^3 + 2 y=14+2y = 1 - 4 + 2 y=1y = -1

So, the point of tangency is (1,1)(1, -1).

Now we can use the point-slope form of the equation of a line:

yy1=m(xx1)y - y_1 = m(x - x_1)

where (x1,y1)(x_1, y_1) is the point of tangency (1,1)(1, -1) and mm is the slope 8-8.

y(1)=8(x1)y - (-1) = -8(x - 1) y+1=8(x1)y + 1 = -8(x - 1) y+1=8x+8y + 1 = -8x + 8 y=8x+81y = -8x + 8 - 1 y=8x+7y = -8x + 7

Therefore, the equation of the tangent line in slope-intercept form is:

y=8x+7y = -8x + 7

Would you like further details or have any questions?

Here are 5 related questions you might want to ask:

  1. How do you convert from point-slope form to slope-intercept form?
  2. What is the significance of the y-intercept in the equation of a line?
  3. Can you explain how to find the equation of a tangent line for other types of functions?
  4. How does the slope of a tangent line relate to the derivative of the function?
  5. What are some applications of finding tangent lines in real-world problems?

Tip: When finding the equation of a tangent line, always ensure that you correctly identify the slope and the coordinates of the point of tangency.

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Math Problem Analysis

Mathematical Concepts

Equation of a Line
Point-Slope Form
Slope-Intercept Form

Formulas

Point-Slope Form
Slope-Intercept Form

Theorems

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Suitable Grade Level

Grades 9-12