Find the Vertical and Horizontal Intercept(s) f(x)=-0.1x^2+0.7x+6

$f (x) = - 0.1 x^{2} + 0.7 x + 6$

Step 1

Find the x-intercepts.

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 0.1 x^{2} + 0.7 x + 6$

Solve the equation.

Rewrite the equation as $- 0.1 x^{2} + 0.7 x + 6 = 0$ .

$- 0.1 x^{2} + 0.7 x + 6 = 0$

Factor the left side of the equation.

Factor $- 0.1$ out of $- 0.1 x^{2} + 0.7 x + 6$ .

Factor $- 0.1$ out of $- 0.1 x^{2}$ .

$- 0.1 x^{2} + 0.7 x + 6 = 0$

Factor $- 0.1$ out of $0.7 x$ .

$- 0.1 x^{2} - 0.1 (- 7 x) + 6 = 0$

Factor $- 0.1$ out of $6$ .

$- 0.1 x^{2} - 0.1 (- 7 x) - 0.1 \cdot - 60 = 0$

Factor $- 0.1$ out of $- 0.1 (x^{2}) - 0.1 (- 7 x)$ .

$- 0.1 (x^{2} - 7 x) - 0.1 \cdot - 60 = 0$

Factor $- 0.1$ out of $- 0.1 (x^{2} - 7 x) - 0.1 (- 60)$ .

$- 0.1 (x^{2} - 7 x - 60) = 0$

Factor.

Factor $x^{2} - 7 x - 60$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 60$ and whose sum is $- 7$ .

$- 12, 5$

Write the factored form using these integers.

$- 0.1 ((x - 12) (x + 5)) = 0$

Remove unnecessary parentheses.

$- 0.1 (x - 12) (x + 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 12 = 0$

$x + 5 = 0$

Set $x - 12$ equal to $0$ and solve for $x$ .

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

The final solution is all the values that make $- 0.1 (x - 12) (x + 5) = 0$ true.

$x = 12, - 5$

x-intercept(s) in point form.

x-intercept(s): $(12, 0), (- 5, 0)$

Step 2

Find the y-intercepts.

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 0.1 {(0)}^{2} + 0.7 (0) + 6$

Solve the equation.

Remove parentheses.

$y = - 0.1 \cdot 0^{2} + 0.7 (0) + 6$

Remove parentheses.

$y = - 0.1 {(0)}^{2} + 0.7 (0) + 6$

Simplify $- 0.1 {(0)}^{2} + 0.7 (0) + 6$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$y = - 0.1 \cdot 0 + 0.7 (0) + 6$

Multiply $- 0.1$ by $0$ .

$y = 0 + 0.7 (0) + 6$

Multiply $0.7$ by $0$ .

$y = 0 + 0 + 6$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 6$

Add $0$ and $6$ .

$y = 6$

y-intercept(s) in point form.

y-intercept(s): $(0, 6)$

Step 3

List the intersections.

x-intercept(s): $(12, 0), (- 5, 0)$

y-intercept(s): $(0, 6)$

Step 4

Find the Vertical and Horizontal Intercept(s) f(x)=-0.1x^2+0.7x+6

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