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In any right triangle, whereaandbare the lengths of the legs, andcis the length of the hypotenuse,a2+b2=c2. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Projectile motionThe height in feet,h, of an object shot upwards into the air with initial velocity,v0, aftertseconds is given by the formulah= â16t2+v0t. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
The height in feet,h, of an object shot upwards into the air with initial velocity,v0, aftertseconds is given by the formulah= â16t2+v0t. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Parabola OrientationFor the graph of the quadratic functionf(x)=ax2+bx+c,f(x)=ax2+bx+c,ifa> 0, the parabola opens upward.a< 0, the parabola opens downward. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
For the graph of the quadratic functionf(x)=ax2+bx+c,f(x)=ax2+bx+c,ifa> 0, the parabola opens upward.a< 0, the parabola opens downward. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
a> 0, the parabola opens upward. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
a< 0, the parabola opens downward. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Axis of Symmetry and Vertex of a Parabola The graph of the functionf(x)=ax2+bx+cf(x)=ax2+bx+cis a parabola where:the axis of symmetry is the vertical linex=âb2a.x=âb2a.the vertex is a point on the axis of symmetry, so itsx-coordinate isâb2a.âb2a.they-coordinate of the vertex is found by substitutingx=âb2ax=â... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
the axis of symmetry is the vertical linex=âb2a.x=âb2a. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
the vertex is a point on the axis of symmetry, so itsx-coordinate isâb2a.âb2a. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
they-coordinate of the vertex is found by substitutingx=âb2ax=âb2ainto the quadratic equation. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Find the Intercepts of a ParabolaTo find the intercepts of a parabola whose function isf(x)=ax2+bx+c:f(x)=ax2+bx+c:y-interceptx-interceptsLetx=0and solve forf(x).Letf(x)=0and solve forx.y-interceptx-interceptsLetx=0and solve forf(x).Letf(x)=0and solve forx. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
To find the intercepts of a parabola whose function isf(x)=ax2+bx+c:f(x)=ax2+bx+c:y-interceptx-interceptsLetx=0and solve forf(x).Letf(x)=0and solve forx.y-interceptx-interceptsLetx=0and solve forf(x).Letf(x)=0and solve forx. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to graph a quadratic function using properties.Step 1.Determine whether the parabola opens upward or downward.Step 2.Find the equation of the axis of symmetry.Step 3.Find the vertex.Step 4.Find they-intercept. Find the point symmetric to they-intercept across the axis of symmetry.Step 5.Find thex-intercepts. Find a... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Determine whether the parabola opens upward or downward. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Find the equation of the axis of symmetry. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Find the vertex. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Find they-intercept. Find the point symmetric to they-intercept across the axis of symmetry. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 5.Find thex-intercepts. Find additional points if needed. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 6.Graph the parabola. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Minimum or Maximum Values of a Quadratic EquationThey-coordinate of the vertex of the graph of a quadratic equation is theminimumvalue of the quadratic equation if the parabola opensupward.maximumvalue of the quadratic equation if the parabola opensdownward. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
They-coordinate of the vertex of the graph of a quadratic equation is the | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
minimumvalue of the quadratic equation if the parabola opensupward. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
maximumvalue of the quadratic equation if the parabola opensdownward. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Graph a Quadratic Function of the formf(x)=x2+kf(x)=x2+kUsing a Vertical ShiftThe graph off(x)=x2+kf(x)=x2+kshifts the graph off(x)=x2f(x)=x2vertically k units.Ifk> 0, shift the parabola vertically upkunits.Ifk< 0, shift the parabola vertically down|k||k|units. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
The graph off(x)=x2+kf(x)=x2+kshifts the graph off(x)=x2f(x)=x2vertically k units.Ifk> 0, shift the parabola vertically upkunits.Ifk< 0, shift the parabola vertically down|k||k|units. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Ifk> 0, shift the parabola vertically upkunits. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Ifk< 0, shift the parabola vertically down|k||k|units. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Graph a Quadratic Function of the formf(x)=(xâh)2f(x)=(xâh)2Using a Horizontal ShiftThe graph off(x)=(xâh)2f(x)=(xâh)2shifts the graph off(x)=x2f(x)=x2horizontally h units.Ifh> 0, shift the parabola horizontally lefthunits.Ifh< 0, shift the parabola horizontally right|h||h|units. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
The graph off(x)=(xâh)2f(x)=(xâh)2shifts the graph off(x)=x2f(x)=x2horizontally h units.Ifh> 0, shift the parabola horizontally lefthunits.Ifh< 0, shift the parabola horizontally right|h||h|units. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Ifh> 0, shift the parabola horizontally lefthunits. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Ifh< 0, shift the parabola horizontally right|h||h|units. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Graph of a Quadratic Function of the formf(x)=ax2f(x)=ax2The coefficientain the functionf(x)=ax2f(x)=ax2affects the graph off(x)=x2f(x)=x2by stretching or compressing it.If0<|a|<1,0<|a|<1,then the graph off(x)=ax2f(x)=ax2will be âwiderâ than the graph off(x)=x2.f(x)=x2.If|a|>1,|a|>1,then the graph off(x)=ax2f(x)=ax... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
The coefficientain the functionf(x)=ax2f(x)=ax2affects the graph off(x)=x2f(x)=x2by stretching or compressing it.If0<|a|<1,0<|a|<1,then the graph off(x)=ax2f(x)=ax2will be âwiderâ than the graph off(x)=x2.f(x)=x2.If|a|>1,|a|>1,then the graph off(x)=ax2f(x)=ax2will be âskinnierâ than the graph off(x)=x2.f(x)=x2. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to graph a quadratic function using transformationsStep 1.Rewrite the function inf(x)=a(xâh)2+kf(x)=a(xâh)2+kform by completing the square.Step 2.Graph the function using transformations. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Rewrite the function inf(x)=a(xâh)2+kf(x)=a(xâh)2+kform by completing the square. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Graph the function using transformations. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Graph a quadratic function in the vertex formf(x)=a(xâh)2+kf(x)=a(xâh)2+kusing propertiesStep 1.Rewrite the function inf(x)=a(xâh)2+kf(x)=a(xâh)2+kform.Step 2.Determine whether the parabola opens upward,a> 0, or downward, a < 0.Step 3.Find the axis of symmetry,x=h.Step 4.Find the vertex, (h,k).Step 5.Find they-... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Rewrite the function inf(x)=a(xâh)2+kf(x)=a(xâh)2+kform. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Determine whether the parabola opens upward,a> 0, or downward, a < 0. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Find the axis of symmetry,x=h. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Find the vertex, (h,k). | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 5.Find they-intercept. Find the point symmetric to they-intercept across the axis of symmetry. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 6.Find thex-intercepts, if possible. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 7.Graph the parabola. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Solve a Quadratic Inequality GraphicallyStep 1.Write the quadratic inequality in standard form.Step 2.Graph the functionf(x)=ax2+bx+cf(x)=ax2+bx+cusing properties or transformations.Step 3.Determine the solution from the graph. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Write the quadratic inequality in standard form. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Graph the functionf(x)=ax2+bx+cf(x)=ax2+bx+cusing properties or transformations. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Determine the solution from the graph. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
How to Solve a Quadratic Inequality AlgebraicallyStep 1.Write the quadratic inequality in standard form.Step 2.Determine the critical points -- the solutions to the related quadratic equation.Step 3.Use the critical points to divide the number line into intervals.Step 4.Above the number line show the sign of each quadr... | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 1.Write the quadratic inequality in standard form. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 2.Determine the critical points -- the solutions to the related quadratic equation. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 3.Use the critical points to divide the number line into intervals. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 4.Above the number line show the sign of each quadratic expression using test points from each interval substituted into the original inequality. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
Step 5.Determine the intervals where the inequality is correct. Write the solution in interval notation. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-concepts |
discriminant : In the Quadratic Formula,x=âb±b2â4ac2a,x=âb±b2â4ac2a,the quantityb2â 4acis called the discriminant. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-terms |
quadratic function : A quadratic function, wherea,b, andcare real numbers andaâ0,aâ0,is a function of the formf(x)=ax2+bx+c.f(x)=ax2+bx+c. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-terms |
quadratic inequality : A quadratic inequality is an inequality that contains a quadratic expression. | https://openstax.org/books/intermediate-algebra-2e/pages/9-key-terms |
x x -axis, the line y = 0 y = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
x x -axis, the line y = 0 y = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
log a 1 = 0 log a 1 = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
ln 1 = 0 ln 1 = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
log a a = 1 log a a = 1 | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
ln e = 1 ln e = 1 | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
a log a x = x log a a x = x a log a x = x log a a x = x | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
e ln x = x ln e x = x e ln x = x ln e x = x | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
log a ( M · N ) = log a M + log a N log a ( M · N ) = log a M + log a N | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
ln ( M · N ) = ln M + ln N ln ( M · N ) = ln M + ln N | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
log a M N = log a M â log a N log a M N = log a M â log a N | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
ln M N = ln M â ln N ln M N = ln M â ln N | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
log a M p = p log a M log a M p = p log a M | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
ln M p = p ln M ln M p = p ln M | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-concepts |
asymptote : A line which a graph of a function approaches closely but never touches. | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-terms |
common logarithmic function : The functionf(x)=logxf(x)=logxis the common logarithmic function with base10,10,wherex>0.x>0.y=logxis equivalent tox=10yy=logxis equivalent tox=10y | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-terms |
exponential function : An exponential function, wherea>0a>0andaâ1,aâ1,is a function of the formf(x)=ax.f(x)=ax. | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-terms |
logarithmic function : The functionf(x)=logaxf(x)=logaxis the logarithmic function with basea,a,wherea>0,a>0,x>0,x>0,andaâ1.aâ1.y=logaxis equivalent tox=ayy=logaxis equivalent tox=ay | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-terms |
natural base : The numbereis defined as the value of(1+1n)n,(1+1n)n,asngets larger and larger. We say, asnincreases without bound,eâ2.718281827...eâ2.718281827... | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-terms |
natural exponential function : The natural exponential function is an exponential function whose base ise:f(x)=ex.f(x)=ex.The domain is(ââ,â)(ââ,â)and the range is(0,â).(0,â). | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-terms |
natural logarithmic function : The functionf(x)=lnxf(x)=lnxis the natural logarithmic function with basee,e,wherex>0.x>0.y=lnxis equivalent tox=eyy=lnxis equivalent tox=ey | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-terms |
one-to-one function : A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, eachy-value is matched with only onex-value. | https://openstax.org/books/intermediate-algebra-2e/pages/10-key-terms |
x = â b 2 a x = â b 2 a | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
x = h x = h | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Substitute x = â b 2 a x = â b 2 a and solve for y . | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Let x = 0 x = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Let x = 0 x = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Let y = 0 y = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Let y = 0 y = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
y = â b 2 a y = â b 2 a | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
y = k y = k | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Substitute y = â b 2 a y = â b 2 a and solve for x . | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Let x = 0 x = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Let x = 0 x = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Let y = 0 y = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Let y = 0 y = 0 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
y = b a x , y = b a x , y = â b a x y = â b a x | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
y = a b x , y = a b x , y = â a b x y = â a b x | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Use a units left/right of center b units above/below the center | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
Use a units above/below the center b units left/right of center | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
x = 3 y 2 â 2 y + 1 x = 3 y 2 â 2 y + 1 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
x 2 - x 2 - and y 2 - y 2 - terms must have the same coefficients and they must be the same sign as the constant after the = sign | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
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