tell whether the graph opens upward or downward. then find the axis of symmetry and vertex of the graph of the function
1. y=xsquared-5
2. y=-2*xsquared+6x+7
x^2 = (y + 5)
when x gets big, y gets big, opens up (holds water)
symmetric about y axis (x = 0)
vertex at (0,-5)
2 x^2 - 6 x = - y + 7
x^2 - 3 x = -(1/2)y + 7/2
add (3/2)^2 = 9/4 to both sides
x^2 -3 x +9/4 = -(2/4)y + 23/4
(x-3/2)^2 = -(1/4)(2y-23)
axis at x = 3/2
vertex at (3/2 , 23/2)
when x gets big, y gets big negative. Opens down (sheds water)
To determine whether the graph opens upward or downward, we need to look at the coefficient of the x^2 term in each equation.
1. For the equation y = x^2 - 5, the coefficient of the x^2 term is 1, which is positive. Since the coefficient is positive, the graph opens upward.
2. For the equation y = -2x^2 + 6x + 7, the coefficient of the x^2 term is -2, which is negative. Therefore, the graph opens downward.
Now, let's find the axis of symmetry and vertex of each graph:
1. To find the axis of symmetry, we can use the formula x = -b/2a, where a and b are the coefficients of the x^2 and x terms, respectively. In this case, a = 1 and b = 0. Plugging these values into the formula, we get x = -0/2(1), which simplifies to x = 0. Therefore, the axis of symmetry for the graph of y = x^2 - 5 is x = 0.
To find the vertex, substitute the x-value of the axis of symmetry, which is 0, into the equation to find the corresponding y-value. Plugging x = 0 into y = x^2 - 5, we get y = 0^2 - 5, which simplifies to y = -5. Therefore, the vertex of the graph is (0, -5).
2. To find the axis of symmetry for the equation y = -2x^2 + 6x + 7, we again use the formula x = -b/2a. Here, a = -2 and b = 6. Substituting these values gives x = -6/2(-2), which simplifies to x = -6/-4. Simplifying further, we find that x = 3/2 or 1.5. Therefore, the axis of symmetry for the graph of y = -2x^2 + 6x + 7 is x = 1.5.
To find the vertex, substitute the x-value of the axis of symmetry, which is 1.5, into the equation to find the corresponding y-value. Plugging x = 1.5 into y = -2x^2 + 6x + 7, we get y = -2(1.5)^2 + 6(1.5) + 7. Evaluating this expression, we find y = -4.5 + 9 + 7, which simplifies to y = 11.5. Therefore, the vertex of the graph is (1.5, 11.5).