Q: How could the function y=3t^2 +4 be plotted on a cartesian graph to produce a straight line? What would be the numerical values of the slope and the intercept of the line?
For this it's explained in the lab book for physics as if the function is y=mx+b it would be a straight line but if it's y=mx^2+b then it is a parabola. It's explained that to get a straight line I should go and make the equation with the square y=mx'+b with x'= x^2. Then it says that if you plotted these numbers with the corresponding y values you'd get a straight line.
I'm not sure I understand how this would work.
For the 2nd part, wouldn't the slope be the same as the original equation (3)? and the y intercept be the same as the original equation too (4)?
Thanks =D
Huh? Just label the x-axis as x^2 instead of x.
Then, when x^2 = 0, y=4
when x^2 = 1, y=7
and so on.
The line plotted will be straight.
This is just like using semi-log graph paper (see google), only the horizontal axis is marked in units of x^2 rather than logx.
To plot the function y=3t^2 +4 on a Cartesian graph as a straight line, you can follow the given explanation in the lab book. By introducing a new variable x' = t^2, you can rewrite the equation as y = 3x' + 4.
When you plot the values of x' on the x-axis and the corresponding values of y on the y-axis, you will indeed get a straight line. This is because the new equation is in the form y = mx' + b, where m is the slope and b is the y-intercept.
In this case, the slope is 3, which means that for every unit increase in x', y increases by 3 units. The y-intercept remains the same as the original equation, which is 4. This means that the line intersects the y-axis at the point (0, 4).
So, to summarize:
- The straight line equation in terms of x' is y = 3x' + 4.
- The slope is 3, indicating that y increases by 3 units for every unit increase in x'.
- The y-intercept is 4, indicating that the line intersects the y-axis at the point (0, 4).
I hope this clarifies the concept for you! Let me know if you have any further questions.
To plot the function y = 3t^2 + 4 on a Cartesian graph as a straight line, you can make a transformation by substituting a new variable x' = t^2 into the equation. This allows us to rewrite the equation as y = mx' + b, where m is the slope and b is the y-intercept.
By substituting x' = t^2 into the equation y = 3t^2 + 4, we have y = 3x' + 4. Now, if we plot the points (x', y) on a graph, we will obtain a straight line.
To find the numerical values of the slope (m) and the y-intercept (b) for this straight line, we can compare it to the standard equation y = mx + b.
In our case, y = 3x' + 4. Therefore, the slope (m) is 3, which means that for every unit increase in x', the value of y will increase by 3. The y-intercept (b) is 4, indicating that the line intersects the y-axis at the point (0, 4).
So, the numerical values of the slope and the y-intercept for the line obtained by plotting the function y = 3t^2 + 4 on a Cartesian graph as a straight line are: slope (m) = 3 and y-intercept (b) = 4.