Graphing Quadratics

x-intercepts: (−10, 0) and (2, 0); y-intercept: (0, −5)

x-intercepts have a y-coordinate of 0: $(-10, 0)$ and $(2, 0)$. The y-intercept has an x-coordinate of 0: $(0, -5)$.

x-intercept: (5, 0); y-intercepts: (0, 10) and (0, −2): The x- and y-intercepts were switched and their signs reversed.

x-intercepts: (0, −10) and (0, 2); y-intercept: (−5, 0): The coordinates in each ordered pair were switched. x-intercepts look like $(x, 0)$.

x-intercept: (0, 5); y-intercepts: (10, 0) and (−2, 0): The intercepts were switched and the signs reversed.

I, II, and III

The graph touches the x-axis only at $(-3, 0)$, crosses the y-axis at $(0, 9)$, and is symmetric about the vertical line $x = -3$. All three statements are true.

I and II only: The axis of symmetry passes through the vertex at $x = -3$, so III is also true.

I and III only: The graph crosses the y-axis at $(0, 9)$, so II is also true.

II and III only: The vertex $(-3, 0)$ sits on the x-axis, so it is the x-intercept. I is also true.

Graph A

The coefficient of $x^2$ is negative, so the parabola opens down. Its axis is $x = -\frac{6}{2(-1)} = 3$, and $y(3)=8$, giving vertex $(3,8)$ and y-intercept −1.

Graph B: This parabola opens upward; the coefficient of $x^2$ is −1.

Graph C: This parabola opens upward rather than downward.

Graph D: Check the y-intercept and zeros against $y(0)=-1$ and the vertex $(3,8)$.

Graph F

The zeros are $x=-1$ and $x=3$. Expanding gives $x^2-2x-3$, so the graph opens upward and has y-intercept −3.

Graph G: This graph does not have both x-intercepts −1 and 3.

Graph H: The vertex and y-intercept do not match $h(x)=x^2-2x-3$.

Graph J: Check the zeros: $x+1=0$ gives −1 and $x-3=0$ gives 3.

Graph A

Exactly one zero means the vertex lies on the x-axis. The axis $x=-3$ places that vertex at $(-3,0)$, as shown in Graph A.

Graph B: This graph has two x-intercepts, not exactly one zero.

Graph C: The vertex is not on the x-axis, so the function has no real zero.

Graph D: This graph has two zeros rather than one.

225

The maximum value is the y-coordinate of the vertex. The vertex is halfway between $t=1.5$ and $t=6$, at $t=3.75$, where the height is 225 feet.

144: 144 is the height at $t=1.5$ and $t=6$, not the maximum.

7.5: 7.5 is the time when the rocket returns to the ground, not a height.

The point at $(0, -3)$

The $y$-intercept is the point where the graph crosses the $y$-axis. This graph crosses the $y$-axis at $(0, -3)$.

$x = -1$ and $x = 3$

Zeros are the x-values where the function has a y-value of zero. The released graph crosses the x-axis at $x = -1$ and $x = 3$.

$x = 1$: This is the x-coordinate of the vertex, not an x-intercept.

$x = -4$: This is the minimum y-value, not a zero of the function.

$x = 0$: At x = 0 the curve is below the x-axis, so its y-value is not zero.

$x = -3$: This is the y-intercept's value, not an x-intercept.

The function has a minimum value of −4.

The graph opens upward and its vertex is $(-1, -4)$, so the function has a minimum value of $-4$.

$x = -5$

The vertex of the graph is $(-5, -7)$. The axis of symmetry is the vertical line $x = h$ through the vertex, so it is $x = -5$.

$x = -3$: The labeled point $(-3, -3)$ has equal coordinates, but it is not the vertex.

$y = -7$: This uses the vertex's $y$-coordinate. The axis of symmetry is a vertical line, $x = h$.

$y = 2$: The labeled points $(-8, 2)$ and $(-2, 2)$ share a $y$-coordinate, but the axis of symmetry is the vertical line halfway between them.

Graph A

The vertex's $x$-coordinate is $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$, and $f(2) = 4 - 8 + 3 = -1$. Graph A is the parabola with vertex $(2, -1)$.

Graph B: This parabola has vertex $(1, 2)$: $f(2)$ was evaluated as 1, and the vertex coordinates were reversed.

Graph C: This parabola has vertex $(-2, -1)$; the sign of the vertex's $x$-coordinate is wrong.

Graph D: This parabola has vertex $(-1, 2)$; the vertex's coordinates were reversed.

$x = 30$

The vertex of the graph is $(30, 50)$. The axis of symmetry is the vertical line through the vertex, $x = 30$.

$y = 50$: This uses the vertex's $y$-coordinate. The axis of symmetry is a vertical line.

$x = 60$: This vertical line passes through the zero at $(60, 0)$, not the vertex.

$y = 25$: The axis of symmetry is a vertical line halfway between the zeros, not a horizontal line.