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## Section10.6Additional Practice Related to Parabolas

### ExercisesExercises

For each equation, state the vertex, axis of symmetry, concavity, and all intercepts of the parabola.

###### 1.

$y=x^2+4x-12$

Solution

The vertex of the parabola is $(-2,16)\text{.}$ The axis of symmetry is the line with equation $x=-2\text{.}$ The parabola is concave up. The $x$-intercepts are $(-6,0)$ and $(2,0)\text{.}$ The $y$-intercept is $(0,-12)\text{.}$

###### 2.

$y=-x^2-3$

Solution

The vertex of the parabola is $(0,-3)\text{.}$ The axis of symmetry is the line with equation $x=0$ (i.e., the $y$-axis). The parabola is concave down. The $y$-intercept is $(0,-3)\text{.}$ The parabola has no $x$-intercepts.

###### 3.

$y=4x^2+8x$

Solution

The vertex of the parabola is $(-1,-4)\text{.}$ The axis of symmetry is the line with equation $x=-1\text{.}$ The parabola is concave up. The $x$-intercepts are $(-2,0)$ and $(0,0)\text{.}$ The $y$-intercept is $(0,0)\text{.}$

###### 4.

$y=\frac{1}{2}x^2-4x+13$

Solution

The vertex of the parabola is $(4,5)\text{.}$ The axis of symmetry is the line with equation $x=4\text{.}$ The parabola is concave up. The $y$-intercept is $(0,13)\text{.}$ The parabola has no $x$-intercepts.

###### 5.

$y=-2x^2+7x-4$

Solution

The vertex of the parabola is $\left(\frac{7}{4},\frac{17}{8}\right)\text{.}$ The axis of symmetry is the line with equation $x=\frac{7}{4}\text{.}$ The parabola is concave down. The $x$-intercepts are $\left(\frac{7-\sqrt{17}}{4},0\right)$ and $\left(\frac{7+\sqrt{17}}{4},0\right)\text{.}$ The $y$-intercept is $(0,-4)\text{.}$

An out-of-the-ox question for you.

###### 6.

The points given in Figure 10.6.1 all lie on the same parabola. Determine the axis-of-symmetry for the parabola and complete the missing entries in the table.

Solution

The axis of symmetry is the line with equation $x=1.5\text{.}$

For each equation, state the vertex, axis of symmetry, concavity, and all intercepts of the parabola.

###### 7.

$y=-(x+4)^2+9$

Solution

The vertex of the parabola is $(-4,9)\text{.}$ The axis of symmetry is the line with equation $x=-4\text{.}$ The parabola is concave down. The $x$-intercepts are $(-1,0)$ and $(-7,0)\text{.}$ The $y$-intercept is $(0,9)\text{.}$

###### 8.

$y=3x^2-12$

Solution

The vertex of the parabola is $(0,-12)\text{.}$ The axis of symmetry is the line with equation $x=0$ (i.e. the $y$-axis). The parabola is concave up. The $x$-intercepts are $(-2,0)$ and $(2,0)\text{.}$ The $y$-intercept is $(0,-12)\text{.}$

###### 9.

$y=2(x-5)^2+16$

Solution

The vertex of the parabola is $(5,16)\text{.}$ The axis of symmetry is the line with equation $x=5\text{.}$ The parabola is concave up. The $y$-intercept is $(0,16)\text{.}$ The parabola has no $x$-intercepts.

###### 10.

$y=-2(x-7)+40$

Solution

The vertex of the parabola is $(7,40)\text{.}$ The axis of symmetry is the line with equation $x=7\text{.}$ The parabola is concave down. The $x$-intercepts are $(7+2\sqrt{5},0)$ and $(7-2\sqrt{5},0)\text{.}$ The $y$-intercept is $(0,40)\text{.}$

###### 11.

$y=-6(x+4)^2$

Solution

The vertex of the parabola is $(-4,0)\text{.}$ The axis of symmetry is the line with equation $x=-4\text{.}$ The parabola is concave down. The only $x$-intercept is $(-4,0)\text{.}$ The $y$-intercept is $(0,-96)\text{.}$

For each graph, determine an equation that would produce the parabola. State the equation in both graphing form and standard form.

###### 12.

Solution

The graphing form of the equation of the parabola shown in Figure 10.6.3 is $y=-2(x+2)^2+6$ and the standard form of the equation is $y=-2x^2-8x-2\text{.}$

###### 13.

Solution

Both the graphing form and standard form of the equation of the parabola shown in Figure 10.6.4 is $y=1.5x^2-3\text{.}$

###### 14.

Solution

The graphing form of the equation of the parabola shown in Figure 10.6.5 is $y=\frac{1}{4}(x-4)^2$ and the standard form of the equation is $y=\frac{1}{4}x^2-2x+4\text{.}$

Determine the vertex of each parabola after first completing the square to determine the graphing form of the equation.

###### 15.

$y=x^2-8x+7$

Solution

The graphing form of the equation is $y=(x-4)^2-9\text{.}$ The vertex of the parabola is $(4,-9)\text{.}$

###### 16.

$y=x^2-5x-2$

Solution

The graphing form of the equation is $y=\left(x-\frac{5}{2}\right)^2-\frac{33}{4}\text{.}$ The vertex of the parabola is $\left(\frac{5}{2},-\frac{33}{4}\right)\text{.}$

###### 17.

$y=2x^2+12x+20$

Solution

The graphing form of the equation is $y=2(x+3)^2+2\text{.}$ The vertex of the parabola is $(-3,2)\text{.}$

###### 18.

$y=-3x^2-24x-15$

Solution

The graphing form of the equation is $y=-3(x+4)^2+33\text{.}$ The vertex of the parabola is $(-4,33)\text{.}$

###### 19.

$y=-2x^2+11x$

Solution

The graphing form of the equation is $y=-2\left(x-\frac{11}{4}\right)^2+\frac{121}{8}\text{.}$ The vertex of the parabola is $\left(\frac{11}{4},\frac{121}{8}\right)\text{.}$

###### 20.

$y=-x^2+10x-28$

Solution

The graphing form of the equation is $y=-(x-5)^2-3\text{.}$ The vertex of the parabola is $(5,-3)\text{.}$