Choose the three true statements about the graph of the quadratic function y = x2 − 3x − 4.
Options
A) The graph is a parabola with a minimum point.
B) The graph is a parabola with a maximum point.
C) The point (2, 2) lies on the graph.
D) The point (1, 6) lies on the graph.
E) The graph crosses the x-axis at x = −4and x =1.
F) The graph crosses the x-axis at x = −1and x =4.
G) The graph crosses the y-axis at y = −4.
because the sign of the second degree term of the polynomial (x^2) is positive, its graph is a parabola with a minimum.
So A is true.
if you fill in -1 and 4 in the equation, you get 0, ergo the graph crosses the x-axis at -1 and 4. So F is true.
For x = 0, y= -4, so the graph crosses the y-axis at -4, so G is also true.
The three true statements about the graph of the quadratic function y = x^2 − 3x − 4 are:
A) The graph is a parabola with a minimum point.
C) The point (2, 2) lies on the graph.
E) The graph crosses the x-axis at x = −4 and x = 1.
To determine which statements are true about the graph of the quadratic function y = x^2 - 3x - 4, we can analyze the equation and the properties of quadratic functions.
1) A) The graph is a parabola with a minimum point.
To determine if the graph has a minimum or maximum point, we look at the coefficient of the x^2 term. In the given equation, the coefficient is positive (+1), which means the parabola opens upwards. Therefore, the statement A) is true.
2) B) The graph is a parabola with a maximum point.
Since the coefficient of the x^2 term is positive (+1) and the parabola opens upwards, it cannot have a maximum point. Therefore, the statement B) is false.
3) C) The point (2, 2) lies on the graph.
To check if a point lies on the graph, we substitute the x and y coordinates into the equation and see if the equation holds true. For this statement, substituting x = 2 and y = 2 into the equation y = x^2 - 3x - 4 gives us:
2 = 2^2 - 3(2) - 4 = 4 - 6 - 4 = -6
Since -6 is not equal to 2, the point (2, 2) does not lie on the graph. Therefore, the statement C) is false.
4) D) The point (1, 6) lies on the graph.
Similar to the previous statement, substituting x = 1 and y = 6 into the equation gives:
6 = 1^2 - 3(1) - 4 = 1 - 3 - 4 = -6
Since -6 is not equal to 6, the point (1, 6) does not lie on the graph. Therefore, the statement D) is false.
5) E) The graph crosses the x-axis at x = -4 and x = 1.
To determine the x-intercepts (where the graph crosses the x-axis), we set y equal to 0 in the equation and solve for x. Setting y = 0, we have:
0 = x^2 - 3x - 4
To find the x-intercepts, we factor the equation or use the quadratic formula. Factoring the equation gives:
0 = (x + 1)(x - 4)
Setting each factor equal to zero, we get:
x + 1 = 0, x = -1
x - 4 = 0, x = 4
Therefore, the graph crosses the x-axis at x = -1 and x = 4. The statement E) is true.
6) F) The graph crosses the x-axis at x = -1 and x = 4.
We already determined the correct x-intercepts in statement E), which are x = -1 and x = 4. Therefore, statement F) is true.
7) G) The graph crosses the y-axis at y = -4.
To find the y-intercept (where the graph crosses the y-axis), we set x equal to 0 in the equation and solve for y. Setting x = 0, we have:
y = (0)^2 - 3(0) - 4 = -4
Therefore, the graph crosses the y-axis at y = -4. The statement G) is true.
In summary, the three true statements are:
A) The graph is a parabola with a minimum point.
E) The graph crosses the x-axis at x = -4 and x = 1.
G) The graph crosses the y-axis at y = -4.