The graph of y=x^2 is a parabola that opens up and has its vertex at the origin (0,0). The graph of y=x is a straight line that passes through the origin and has a slope of 1. These two graphs are related in several ways:
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They intersect at two points
The graphs of y=x^2 and y=x intersect at two points: (0,0) and (1,1). To find these points, we can set the equations equal to each other and solve for x:
y=x^2 y=x x^2=x x(x-1)=0 x=0 or x=1
Then, we can plug these values of x into either equation to find the corresponding values of y:
y=x^2 y=0^2 y=0
y=x^2 y=1^2 y=1
So, the points of intersection are (0,0) and (1,1).
They have the same axis of symmetry
The graphs of y=x^2 and y=x have the same axis of symmetry, which is the vertical line x=0. This means that if we fold the graph along this line, the two halves will match exactly. The axis of symmetry of a parabola can be found by using the formula x=-b/2a, where a and b are the coefficients of x^2 and x in the equation y=ax^2+bx+c. In this case, a=1 and b=0, so x=-b/2a becomes x=-0/2(1), which simplifies to x=0. The axis of symmetry of a line can be found by using the formula x=-c/b, where c and b are the constants in the equation y=mx+b. In this case, c=0 and b=1, so x=-c/b becomes x=-0/1, which simplifies to x=0.
They have different shapes and rates of change
The graphs of y=x^2 and y=x have different shapes and rates of change. The graph of y=x^2 is curved and gets steeper as x increases or decreases. The graph of y=x is linear and has a constant slope of 1. The rate of change of a function can be measured by its derivative, which gives the slope of the tangent line at any point. The derivative of y=x^2 is 2x, which means that the slope of the tangent line depends on the value of x. The derivative of y=x is 1, which means that the slope of the tangent line is always 1.
Here is a graph showing both functions and their derivatives:
According to Desmos
